boundary element method for solving inverse heat …boundary element method for solving inverse heat...

Boundary Element Method for Solving

Inverse Heat Source Problems

Areena Hazanee

Submitted in accordance with the requirements for the degree of

Doctor of Philosophy

The University of Leeds

Department of Applied Mathematics

July 2015

The candidate confirms that the work submitted is his/her own, ex-

cept where work which has formed part of jointly authored publications

has been included. The contribution of the candidate and theother au-

thors to this work has been explicitly indicated below. The candidate

confirms that appropriate credit has been given within the thesis where

reference has been made to the work of others.

This copy has been supplied on the understanding that it is copyright

material and that no quotation from the thesis may be published without

proper acknowledgement.

@2015 The University of Leeds and Areena Hazanee

ii

Joint Publications

Some material of this thesis is based on five published papersand three conference

proceedings as well as one paper pending, as follows:

Publications

1. Hazanee, A., Ismailov, M. I., Lesnic, D. and Kerimov, N. B.(2013) An inverse

time-dependent source problem for the heat equation,Applied Numerical Math-

ematics, Vol.69, pp.13–33.

2. Hazanee, A. and Lesnic, D. (2013) Determination of a time-dependent heat

source from nonlocal boundary conditions,Engineering Analysis with Bound-

ary Elements, Vol.37, No.6, pp.936–956.

3. Hazanee, A. and Lesnic, D. (2013) Reconstruction of an additive space- and

time-dependent heat source,European Journal of Computational Mechanics,

Vol.22, Nos.5–6, pp.304–329.

4. Hazanee, A. and Lesnic, D. (2014) Determination of a time-dependent coef-

ficient in the bioheat equation,International Journal of Mechanical Sciences,

Vol.88, pp.259–266.

5. Hazanee, A., Ismailov, M. I., Lesnic, D. and Kerimov, N. B.(2015) An inverse

time-dependent source problem for the heat equation with a non-classical bound-

ary condition,Applied Mathematical Modelling, in press (available online).

Conference Proceedings

1. Hazanee, A. and Lesnic, D. (2013) The boundary element method for solving an

inverse time-dependent source problem,Proceedings of the 9th UK Conference

on Boundary Integral Methods, (ed. O. Menshykov), UniPrint, University of

Aberdeen, pp.55–62.

iii

2. Hazanee, A. and Lesnic, D. (2014) A time-dependent coefficient identification

problem for the bioheat equation,ICIPE2014 8th International Conference on

Inverse Problems in Engineering, (eds. I. Szczygiel, A. J. Nowak and M. Ro-

jczyk), Silesian University of Technology, May 12-15, 2014, Poland, pp.127–

136.

3. Hazanee, A. and Lesnic, D. (2015) The boundary element method for an in-

verse time-dependent source problem for the heat equation with a non-classical

boundary condition,Proceedings of the 10th UK Conference on Boundary Inte-

gral Methods, (ed. P. J. Harris), University of Brighton, pp.28–35.

Possible Publications

1. Hazanee, A. and Lesnic, D. (2015) Reconstruction of multiplicative space- and

time-dependent sources, submitted to Inverse Problems in Science and Engineer-

ing.

iv

Acknowledgements v

Acknowledgements

In the name of Allah, the most gracious and merciful, Alhamdulillah I thank Al-

lah (subhanahu wa ta’ala) for endowing me with health, patience, and knowledge to

complete this work. This thesis appears in its current form due to the assistance and

guidance of several people. I would like to extend my sincereappreciation to all of

them.

First and foremost, I would like to express my deepest gratitude to my supervisor,

Professor Daniel Lesnic, for providing me with this great opportunity to do my PhD

under his great supervision. My gratitude extends to Dr. EvyKersale, my adviser, for

his valuable advice during my whole PhD study. I also would like to extend my thanks

to all PhD students in the inverse problem group, Bandar Bin-Mohsin, Mohammed

Hussein, Shilan Hussein, Taysir Dyhoum and Dimitra Flouri.

I would like to give my special thanks to all staff in the School of Mathematics

for their help. Many thanks to all my friends in this school, Abeer Al-Nahdi, Wafa

Al-Mohri, Mufida Mohamed Hmaida, and especially to my best friend and sister, Aolo

Bashar Abusaksaka, who was always besides and supported me in any situation I have

faced. I am grateful to Thai students, Joy Riyapan, Jeab Pattamawadee, Jiab Dussadee

for their very good suggestions and providing me the Thai food which made me felt

like I am still in Thailand. And also I am thankful to all Thai Muslim students in the

UK for their friendship and fraternity during my time in the UK.

Of course, the indispensable thanks go to my Hazanee family,dad, mom, brothers,

and my beloved aunt for unconditional love, support, cheering me up and standing by

me. Far distance and time difference could not make me feel I am away from them.

I also gratefully acknowledge the funding support providedby the Ministry of Sci-

ence and Technology of Thailand, Prince of Songkla University Thailand, for pursuing

my PhD at the University of Leeds. I would also like to give special mention to all

staff in the Office of Educational Affairs (OEA) in London fortheir advice and help

dealing with my enquiries.

Abstract vi

Abstract

In this thesis, the boundary element method (BEM) is appliedfor solving inverse

source problems for the heat equation. Through the employment of the Green’s for-

mula and fundamental solution, the BEM naturally reduces the dimensionality of the

problem by one although domain integrals are still present due to the initial condition

and the heat source. We mainly consider the identification oftime-dependent source

for heat equation with several types of conditions such as non-local, non-classical,

periodic, fixed point, time-average and integral which are considered as boundary or

overdetermination conditions. Moreover, the more challenging cases of finding the

space- and time-dependent heat source functions for additive and multiplicative cases

are also considered.

Under the above additional conditions a unique solution is known to exist, however,

the inverse problems are still ill-posed since small errorsin the input measurements re-

sult in large errors in the output heat source solution. Thensome type of regularisation

method is required to stabilise the solution. We utilise regularisation methods such as

the Tikhonov regularisation with order zero, one, two, or the truncated singular value

decomposition (TSVD) together with various choices of the regularisation parameter.

The numerical results obtained from several benchmark testexamples are presented

in order to verify the efficiency of adopted computational methodology. The retrieved

numerical solutions are compared with their analytical solutions, if available, or with

the corresponding direct numerical solution, otherwise. Accurate and stable numeri-

cal solutions have been obtained throughout for all the inverse heat source problems

considered.

Nomenclature vii

Nomenclature

Roman Symbols

A, A0, AL, A(1)k ,A(1)

0,k, A(1)L,k, A

I , AIIi

BEM coefficient matrices

A0j(x, t), ALj(x, t) BEM coefficient functions

B, B∗, B0,BL,B(1)k , B(1)∗

k , B(1)L,k, B

I , BIIi , BIII


B0j(x, t), BLj(x, t) BEM coefficient integral functions

C, C(1)k , CI , CII

i , CIII BEM coefficient matrices

Ck([0, T ]) the space ofk-order continuously differentiable functions

on [0, T ]

Ck,l(DT ) the space ofk-order andl-order continuously differentiable

functions in time and space, respectively

d(x, t), d0(x, t), d1(x, t), d2(x, t)

BEM integral functions for the source term

d¯, d

¯I , d

¯II BEM coefficient vectors

D,D0,D1,D2,D(1)k ,DI

1,DI2,DII

1 ,DII2 ,Dr,Dr

k,DrI ,DrIII


DT := (0, L)× (0, T ) solution domain

DT := [0, L]× [0, T ] closure of the solution domainDT

ei discrete components of functionE(t)

E(t) mass or energy

E¯

vector ofE(t)

E¯ǫ noisy vector ofE(t)

E = diag(ei) diagonal matrix with componentsei

f(x, t), F (x, t), f(x, t) source functions

F, F0, Fλ objective functions

G(x, t, ξ, τ) fundamental solution for one-dimensional heat equation

Nomenclature viii

h(x, t) given source function

h0j , hLj boundary temperatures

h¯, h

¯0, h

¯Lboundary temperature vectors

H(t) Heaviside step function

Hγ/2[0, T ],Hγ[0, L],H2+γ,1+γ/2(DT )

Holder spaces in Chapter 6

i, j, k indices

k(t), ki(t) given boundary and overdetermination functions

k∗ the largest index of spacex satisfyingxk∗ < X0

k¯

vector of functionk(t)

L length of one-dimensional space domain

n outward unit normal to the space boundary

N number of time steps

N0 number of space cells

Nt truncation number (for TSVD)

p, p0 percentages of perturbation

P (t) perfusion coefficient function

q¯, q

¯0, q

¯Lboundary heat flux vectors

q0j , qLj boundary heat fluxes

r(t), r1(t) time-dependent source functions

R, R0, R1, R2, R(1), R(2), Rregularisation matrices

s(x), s1(x) space-dependent source functions

S coefficient matrix

S0, S0 values of the sources at the fixed point

T , T1, T2 final and fixed times

u(x, t), u(x, t) temperatures

u0(x), u0(x) initial temperature

u0,k initial temperature atxk

Nomenclature ix

u¯0

, u¯0

initial temperature vectors

U¯, V

¯orthogonal vectors for SVD

v(x, t) transformation function

v1(t), v2(t) given functions

V1, V2 diagonal matrices ofv1(t), v2(t)

W12(0, T ), W

22(0, T ), W

22(0, L), W

42(0, L), W

4,22 (DT )

Sobolev space in Chapter 7

w¯

unknown vector ofr(t) ands(x)

x, y, x spaces (variable)

X,Xi left-hand side coefficient matrices of BEM linear system

X0 fixed location

Xǫ contaminated left-hand side coefficient matrix

yn eigenfunction of the spectral problem

y¯, y

¯iright-hand side vector of BEM linear system

y¯ǫ noisy right-hand side vector

Greek Symbols

α, a, b heat transfer coefficients

β(x), β1(x), β2(x), β(x, t)

given functions

δij the Kronecker delta symbol

ψ(x) given function

χ(t), χ1(t) given functions

ǫ, ǫ0 noise levels

γ1, γ2 fixed point source values

γij(t) given functions

λ, λL, λdis, λGCV , λopt regularisation parameters

µ1(t), µ2(t), µ0(t), µL(t)

given boundary temperatures

Contents x

Ω domain

∂Ω boundary of the domainΩ

Ω = Ω ∪ ∂Ω closure of the domainΩ

Φ4n0

space of functions

τ , t times (variable)

σ, σψ, σχ, σµ, σr, σs standard deviations

σi singular values

Σ diagonal matrix with components ofσi

ϑ(x, t) given function

Abbreviations

BEM boundary element method

CBEM constant boundary element method

erf, erfc error and complementary error functions

FDM finite difference method

FEM finite element method

FOTR first-order Tikhonov regularisation

GCV generalised cross-validation

PDE partial differential equation

RMSE root mean square error

SOTR second-order Tikhonov regularisation

SVD singular value decomposition

TSVD truncated singular value decomposition

ZOTR zeroth-order Tikhonov regularisation

Contents

Joint Publications iii

Acknowledgements v

Abstract vi

Nomenclature vii

Contents xi

List of Figures xiv

List of Tables xxvi

1 General Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Inverse and ill-posed problems . . . . . . . . . . . . . . . . . . . . .1

1.3 The boundary element method (BEM) . . . . . . . . . . . . . . . . . 2

1.4 The BEM for solving one-dimensional direct heat problem. . . . . . 4

1.5 Condition number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Regularisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6.1 The truncated singular value decomposition (TSVD) . .. . . 9

1.6.2 The Tikhonov regularisation . . . . . . . . . . . . . . . . . . 10

1.6.3 Choice of the regularisation parameter . . . . . . . . . . . .. 11

xi

Contents xii

1.7 Purpose of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Determination of a Time-dependent Heat Source from Nonlocal Boundary

Conditions 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . 18


2.3.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.4 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.5 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.6 Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Numerical examples and discussion . . . . . . . . . . . . . . . . . .31

2.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Determination of a Time-dependent Heat Source from Nonlocal Boundary

and Integral Conditions 61

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61



3.4 Regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66


3.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Contents xiii

4 Determination of a Time-dependent Coefficient in the Bioheat Equation 83

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83



4.4 Regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87


4.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Determination of a Time-dependent Heat Source with a Dynamic Bound-

ary Condition 99

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99


5.3 Boundary element method (BEM) . . . . . . . . . . . . . . . . . . . 103


5.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6 Determination of an Additive Space- and Time-dependent Heat Source 121

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121



6.4 Regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129


6.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

List of Figures xiv

7 Determination of Multiplicative Space- and Time-dependent Heat Sources149

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149



7.4 Solution of inverse problem . . . . . . . . . . . . . . . . . . . . . . . 155


7.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8 General Conclusions and Future Work 177

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Bibliography 185

List of Figures

2.1 The normalised singular values of matrixX for (a) Case 1 – (f) Case 6

for N0 = N = 20 (− · −), 40 (· · · ), 80 (−−−), for Example 1. . . 33

2.2 The analytical (—–) and numerical (− · −) results of (a)r(t), (b)

u(0, t), and (c)u(1, t) for exact data andλ = 0, for Example 1 Case 1. 35

2.3 The analytical (—–) and numerical (− · −) results ofr(t) obtained by

using the zeroth-order Tikhonov regularisation with the regularisation

parameters (a)λ = 10−7 gives RMSE=0.319, and (b)λ = 10−5 gives

RMSE=0.499, for exact data for Example 1 Case 1. . . . . . . . . . . 36


u(0, t), and (c)u(1, t) for p = 1% noisy data andλ = 0, for Example

1 Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 (a) The discrepancy principle curve, and the analytical(—–) and nu-

merical results of (b)r(t), (c) u(0, t), and (d)u(1, t) obtained using

the zeroth-order Tikhonov regularisation forp = 1% (− · −) and

p = 3% (− − −) noise with the regularisation parametersλdis given

in Table 2.2, for Example 1 Case 1. . . . . . . . . . . . . . . . . . . . 38


merical results of (b)r(t), (c) u(0, t), and (d)u(1, t) obtained us-

ing the first-order Tikhonov regularisation forp = 1% (− · −) and



xv

List of Figures xvi


u(1, t), and (c)ux(0, t) for exact data andλ = 0, for Example 1 Case 2. 40


u(1, t), and (c)ux(0, t) for p = 1% noisy data andλ = 0, for Ex-

ample 1 Case 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


merical results of (b)r(t), (c) u(1, t), and (d)ux(0, t) obtained us-

ing the zeroth-order Tikhonov regularisation forp = 1% (− · −) and



2.10 (a) The discrepancy principle curve, and the analytical (—–) and nu-

merical results of (b)r(t), (c) u(1, t), and (d)ux(0, t) obtained us-





ux(0, t), and (c)ux(1, t) for p = 1% noisy data andλ = 0, for Ex-

ample 1 Case 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


merical results of (b)r(t), (c) ux(0, t), and (d)ux(1, t) obtained us-

ing the zeroth-order Tikhonov regularisation forp = 1% (− · −) and




merical results of (b)r(t), (c) ux(0, t), and (d)ux(1, t) obtained us-




List of Figures xvii


u(1, t), (c) ux(0, t), andux(1, t) for p = 1% noisy data andλ = 0,

for Example 1 Case 4. . . . . . . . . . . . . . . . . . . . . . . . . . . 47


merical results of (b)r(t), (c) u(1, t), (d) ux(0, t), and (e)ux(1, t) ob-

tained using the zeroth-order Tikhonov regularisation forp = 1% (− ·−) andp = 3% (−−−) noise with the regularisation parametersλdis

given in Table 2.5, for Example 1 Case 4. . . . . . . . . . . . . . . . 49


merical results of (b)r(t), (c) u(1, t), (d) ux(0, t), and (e)ux(1, t) ob-

tained using the first-order Tikhonov regularisation forp = 1% (−·−)

andp = 3% (− − −) noise with the regularisation parametersλdis



u(0, t), (c) u(1, t) and (d)ux(1, t) for exact data andλ = 0, for Exam-

ple 1 Case 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


u(0, t), (c) u(1, t) and (d)ux(1, t) for p = 1% noisy data andλ = 0,

for Example 1 Case 5. . . . . . . . . . . . . . . . . . . . . . . . . . . 52


merical results of (b)r(t), (c) u(0, t), (d) u(1, t), and (e)ux(1, t) ob-

tained using the zeroth-order Tikhonov regularisation forp = 1% (− ·−) andp = 3% (−−−) noise with the regularisation parametersλdis



merical results of (b)r(t), (c) u(0, t), (d) u(1, t), and (e)ux(1, t) ob-

tained using the first-order Tikhonov regularisation forp = 1% (−·−)

andp = 3% (− − −) noise with the regularisation parametersλdis


List of Figures xviii


u(0, t), (c) u(1, t), (d) ux(0, t), and (e)ux(1, t) for exact data and

λ = 0, for Example 1 Case 6. . . . . . . . . . . . . . . . . . . . . . . 55


merical results of (b)r(t), (c) u(0, t), (d) u(1, t), (e)ux(0, t), and (f)

ux(1, t) obtained using the zeroth-order Tikhonov regularisation for

p ∈ 1(− · −), 3(· · · ), 5(− − −)% noise with the regularisation

parametersλdis given in Table 2.7, for Example 1 Case 6. . . . . . . . 56


merical results of (b)r(t), (c) u(0, t), (d) u(1, t), (e)ux(0, t), and (f)

ux(1, t) obtained using the first-order Tikhonov regularisation forp ∈1(− · −), 3(· · · ), 5(−−−)% noise with the regularisation param-

etersλdis given in Table 2.7, for Example 1 Case 6. . . . . . . . . . . 57


merical results of (b)r(t) obtained using the first-order Tikhonov reg-

ularisation forp ∈ 0(), 0.1(−·−), 0.5(· · · ), 1(−−−)% noise

with the regularisation parametersλdis given in Table 2.8, for Example 2. 58

3.1 The normalised singular values of matrixX for N0 = N ∈ 20 (− ·−), 40 (· · · ), 80 (−−−), for Example 1. . . . . . . . . . . . . . . . 68

3.2 The analytical (—–) and numerical (− · −) results of (a)r(t) and (b)

u(0, t) for exact data andλ = 0, for Example 1. . . . . . . . . . . . . 69

3.3 The analytical (—–) and numerical (− · −) results of (a)r(t) and (b)

u(0, t) for p = 1% noisy data andλ = 0, for Example 1. . . . . . . . 70

3.4 TheL-curve on (a), (c), (e) log-log scale and (b), (d), (f) normal

scale, obtained using TSVD forp ∈ 1(top), 3(middle), 5(bottom)%noise, for Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

List of Figures xix


merical results of (b)r(t), and (c)u(0, t) obtained using the TSVD for

p ∈ 1(− · −), 3(· · · ), 5(− − −)% noise at the truncation levelNt

given in Table 3.1, for Example 1. . . . . . . . . . . . . . . . . . . . 73


merical results of (b)r(t), and (c)u(0, t) obtained using the ZOTR for


parametersλdis given in Table 3.1, for Example 1. . . . . . . . . . . . 74


merical results of (b)r(t), and (c)u(0, t) obtained using the FOTR for




merical results of (b)r(t), and (c)u(0, t) obtained using the SOTR for



3.9 The numerical results of (a)E(t) and (b)u(0, t) obtained by solving

the direct problem withN0 = N ∈ 20 (−·−), 40 (· · · ), 80 (−−−),

for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.10 The analytical (—–) and numerical(− · −) results of (a)r(t) and (b)

u(0, t) for exact data andλ = 0, for Example 2. . . . . . . . . . . . . 78


merical results of (b)r(t), and (c)u(0, t) obtained using the ZOTR for




merical results of (b)r(t), and (c)u(0, t) obtained using the FOTR for



List of Figures xx


merical results of (b)r(t), and (c)u(0, t) obtained using the SOTR for



4.1 The analytical (—–) and numerical(− · −) results of (a)r(t), (b)

u(0, t), (c) r′(t), and (d)P (t) obtained using no regularisation, i.e.

λ = 0, for exact data, for Example 1. . . . . . . . . . . . . . . . . . . 90

4.2 The analytical (—–) and numerical(− · −) results of (a)r(t), (b)

u(0, t), (c) r′(t), and (d)P (t) obtained using no regularisation, i.e.

λ = 0, for p = 1% noise, for Example 1. . . . . . . . . . . . . . . . . 91

4.3 The GCV function (4.21), obtained using the second-order Tikhonov

regularisation forp = 1% noise, for Example 1. . . . . . . . . . . . . 92

4.4 The analytical (—–) and numerical results of (a)r(t), (b) u(0, t), (c)

r′(t), and (d)P (t) obtained using the second-order Tikhonov regular-

isation withλGCV = 1.25 × 10−5 (− · −) andλopt = 1.05 × 10−6

( ), for p = 1% noise, for Example 1. . . . . . . . . . . . . . . . 93

4.5 The analytical (—–) and numerical results of (a)r(t), (b) u(0, t)(t),

(c) r′(t), and (d)P (t) obtained using no regularisation in (4.31), i.e.

Λ = 0, for exact data( ) and forp = 1% noisy data(− · −), for

Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.6 The analytical (—–) and numerical results of (a)r′(t) and (b)P (t)

obtained using the smoothing spline regularisation withΛdis = 7 ×10−5 ( ) andΛ = 8 × 10−3 (− · −), as defined in (4.34), for

p = 1% noise, for Example 2. . . . . . . . . . . . . . . . . . . . . . . 97

5.1 The analytical (—–) and numerical results(− · −) of (a) r(t), (b)

u(1, t), (c) ux(0, t), and (d)ux(1, t) for exact data, for Example 1. . . 108

List of Figures xxi


ux(0, t), and (d)ux(1, t) obtained using the straightforward inversion

(− · −) with no regularisation, and the second-order Tikhonov regu-

larisation() with the regularisation parameterλ=4.3E-6 suggested

by the GCV method, forp = 1% noise, for Example 1. . . . . . . . . 109


ux(0, t), and (d)ux(1, t) obtained using the second-order Tikhonov

regularisation with the regularisation parameter suggested by the GCV

method, forp = 3% (· · ·) andp = 5% (−−−), for Example 1. . . . 110

5.4 The numerical results of (a)u(1, t), (b) ux(0, t), (c) ux(1, t), and (d)

E(t) obtained by solving the direct problem withN = N0 ∈ 20( ), 40(· · ·), 80(−−−), for Example 2. . . . . . . . . . . . . . . . 112

5.5 The analytical solution (5.21) and the direct problem numerical solu-

tion from Figures 5.7(a)–5.7(c) (—–) and numerical resultsof (a)r(t),

(b) u(1, t), (c) ux(0, t), and (d)ux(1, t), with no regularisation, for ex-

act data( ) and noisy datap = 1% (− · −), for Example 2. . . . . 114

5.6 The analytical solution (5.21) and the direct problem numerical solu-

tions from Figures 5.7(a)–5.7(c) (—–), and the numerical results of

(a) r(t), (b) u(1, t), (c) ux(0, t), and (d)ux(1, t) obtained using the

second-order Tikhonov regularisation with the regularisation parame-

ters suggested by GCV method, forp ∈ 1(), 3(· · · ), 5(−−−)%noise, for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.7 The numerical results of (a)u(1, t), (b) ux(0, t), (c) ux(1, t), and (d)

E(t) obtained by solving the direct problem withN = N0 ∈ 20 (− ·−), 40 (· · ·), 80 (−−−), for Example 3. . . . . . . . . . . . . . . 117

5.8 The analytical solution (5.4.3) and the direct problem numerical so-

lution from Figures 5.7(a)–5.7(c) (—–), and numerical results of (a)

r(t), (b) u(1, t), (c) ux(0, t), and (d)ux(1, t), with no regularisation,

for exact data( ) and noisy datap = 1% (− · −), for Example 3. 118

List of Figures xxii

5.9 The analytical solution (5.4.3) and the direct problem numerical solu-

tions from Figures 5.7(a)–5.7(c) (—–), and the numerical results of (a)

r(t), (b)u(1, t), (c)ux(0, t), and (d)ux(1, t) obtained using the second-

order Tikhonov regularisation with the regularisation parameters sug-

gested by the discrepancy method, forp ∈ 1(− · −), 3(· · · ), 5(−−−)% noise, for Example 3. . . . . . . . . . . . . . . . . . . . . . . 119

6.1 The normalised singular values of matrixX forN = N0 ∈ 20, 40, 80andX0 ∈ 1

4(− · −), 1

2(· · · ), 3

4(−−−), for Example 1. . . . . . . 132

6.2 The analytical (—–) and numerical results(−·−) of (a)r(t), (b) s(x),

(c) ux(0, t), and (d)ux(1, t) obtained using the SVD for exact data, for

Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3 (a) TheL-curve and (b) the GCV function obtained by the TSVD for

exact data, for Example 1. . . . . . . . . . . . . . . . . . . . . . . . 135

6.4 (a) TheL-curve, and (b) the GCV function, obtained by the ZOTR

(− · −), FOTR (· · · ), and SOTR(− − −) for exact data, withλ =

λr = λs, for Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.5 The analytical (—–) and numerical results of (a)r(t), (b) s(x), (c)

ux(0, t), and (d)ux(1, t) obtained using the TSVD(− + −), ZOTR

(−·−), FOTR(· · · ), and SOTR(−−−) with regularisation parameters

suggested by the GCV function of Figure 6.3(b) and 6.4(b) forexact

data, for Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.6 The analytical (—–) and numerical results(−·−) of (a)r(t), (b) s(x),

(c) ux(0, t), and (d)ux(1, t) obtained using the SOTR with the regular-

isation parameterλL=1E-1 suggested by theL-curve of Figure 6.4(a)

for exact data, for Example 1. . . . . . . . . . . . . . . . . . . . . . . 137

6.7 (a) TheL-curve and (b) the discrepancy principle obtained using the

TSVD for noisy inputp = 1%, for Example 1. . . . . . . . . . . . . . 138

List of Figures xxiii

6.8 (a) TheL-curve and (b) the discrepancy principle obtained using the

ZOTR(−·−), FOTR(· · · ), and SOTR(−−−) for noisy inputp = 1%,

with λ = λr = λs, for Example 1. . . . . . . . . . . . . . . . . . . . 138


ux(0, t), and (d)ux(1, t) obtained using the TSVD(−+−) with Nt =

14, and the ZOTR(− · −), FOTR (· · · ), SOTR(− − −) with regu-

larisation parameters suggested by the discrepancy principle of Figure

6.8(b) for noisy inputp = 1%, for Example 1. . . . . . . . . . . . . . 139

6.10 TheL-surface on (a) a three-dimensional plot, (b) plane oflog ‖Xw¯λ

−y¯ǫ‖ versuslog ‖R(1)r

¯λ‖, and (c) plane oflog ‖Xw

¯λ−y

¯ǫ‖ versuslog ‖R(2)s

¯λ‖,

obtained using the SOTR for noisy inputp = 1%, for Example 1. . . . 140


ux(0, t), and (d)ux(1, t) obtained using the SOTR with regularisation

parameters suggested by theL-curve criterionλL = λr = λs = 10

(−−−), theL-surface method(λr,L, λs,L)=(10,1)(−+−), and the trial

and error(λr,opt, λs,opt)=(8,5.2E-2)(− ∗ −), for noisy inputp = 1%,

for Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.12 The normalised singular values of matrixX forN = N0 = 20 (−·−),

N = N0 = 40 (· · · ), andN = N0 = 80 (−−−), for Example 2. . . . 143

6.13 TheL-curve obtained using (a) the TSVD and (b) the ZOTR(− · −),

FOTR(· · · ), and SOTR(−−−) with λ = λr = λs, for exact data, for

Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144


ux(0, t), and (d)ux(1, t) obtained using the SVD(−·−) and the SOTR

(−o−) with the regularisation parameterλL=1E-4 suggested by theL-

curve of Figure 6.13(b) for exact data, for Example 2. . . . . . .. . . 145

6.15 TheL-curve obtained using (a) the TSVD and (b) the ZOTR(− · −),

FOTR (· · · ), and SOTR(− − −) with λ = λr = λs, for noisy input

p = 1%, for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . 146

List of Figures xxiv

6.16 TheL-surface on (a) a three-dimensional plot, (b) plane oflog ‖Xw¯λ

−y¯ǫ‖ versuslog ‖R(1)r

¯λ‖, and (c) plane oflog ‖Xw

¯λ−y


¯λ‖,

obtained using the SOTR for noisy inputp = 1%, for Example 2. . . . 147


ux(0, t), and (d)ux(1, t) obtained using the SOTR with regularisation

parameters suggested by theL-curve criterionλL = λr = λs = 1

(− − −), theL-surface method(λr,L, λs,L)=(1,10)(− + −), and the

trial and error(λr,opt, λs,opt)=(2.2,5.9)(−∗−), for noisy inputp = 1%,

for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.1 The analytical (—–) and numerical results for (a)χ(t) and (b)β(x)

obtained using the BEM for the direct problem withN = N0 ∈10 (− · −), 20 (· · · ), 40 (−−−), for Example 1. . . . . . . . . . . 158

7.2 (a) The objective functionF0 and the numerical results for (b)r(t), (c)

s(x), (d) u(0, t), (e)u(0.1, t) obtained with no regularisation(− · −),

for exact data for Example 1. The corresponding analytical solutions

are shown by continuous line (—–) in (b)–(e) and thep0 = 100%

perturbed initial guesses are shown by dotted line(· · · ) in (b) and (c). 160

7.3 The numerical results for (a)r(t), (b) s(x), (c) u(0, t), (d) u(0.1, t)

obtained with the first-order regularisation(· · · ) and the second-order

regularisation(−−−) with regularisation parameterλopt = 10−5, for

exact data for Example 1. The corresponding analytical solutions are

shown by continuous line (—–). . . . . . . . . . . . . . . . . . . . . 161

7.4 (a) TheL-curve criterion, (b) the objective functionFλ, and the nu-

merical results(− −) for (c) r(t), (d) s(x), (e) u(0, t), (f) u(0.1, t)

obtained with the hybrid-order regularisation (7.26) withregularisation

parameterλL = 10−5 suggested byL-curve, for exact data Example 1.

The corresponding analytical solutions are shown by continuous line

(—–) in (c)–(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

List of Figures xxv

7.5 (a) The objective functionFλ and the numerical results for (b)r(t), (c)

s(x), (d) u(0, t), (e) u(0.1, t) obtained with the hybrid-order regular-

isation (7.26) with regularisation parameterλL = 10−5 suggested by

L-curve forp ∈ 1(−·−), 3(· · · ), 5(−−−)% noisy data, for Exam-

ple 1. The corresponding analytical solutions are shown by continuous

line (—–) in (b)–(e). . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.6 The analytical (—–) and numerical results for (a)χ(t) and (b)β(x)

obtained using the BEM for the direct problem withN = N0 ∈ 5(−·−), 10(· · · ), 20(−−−), for Example 2. . . . . . . . . . . . . . . 167

7.7 (a) The objective functionF0, (b) the RMSEs forr(t) (−−−) ands(x)

(· · · ) obtained with no regularisation for exact data, and the numerical

results for (c)r(t) and (d)s(x) obtained using the minimisation pro-

cess after 56 unfixed iterations(−·−), and 31 fixed iterations(), for

Example 2. The corresponding analytical solutions (7.30) are shown

by continuous line (—–) in (c) and (d). . . . . . . . . . . . . . . . . 168

7.8 (a) The objective functionFλ, (b) the RMSEs forr(t) (−−−) ands(x)

(· · · ) obtained using the hybrid-order regularisation (7.26) with regu-

larisation parameterλopt = 2× 10−4 for exact data, and the numerical

results for (c)r(t) and (d)s(x) obtained using minimisation process

after 28 unfixed iterations(− · −), and 23 fixed iterations( ), for

Example 2. The corresponding analytical solutions (7.30) are shown

by continuous line (—–) in (c) and (d). . . . . . . . . . . . . . . . . 169

7.9 (a) The objective functionFλ, (b) the RMSEs forr(t) (− − −) and

s(x) (· · · ) obtained using the hybrid-order regularisation (7.26) with

regularisation parameterλopt = 4 × 10−4 for noise levelǫ0 = 0.01,

and the numerical results for (c)r(t) and (d)s(x) obtained using the

minimisation process after 27 unfixed iterations(− · −), and 21 fixed

iterations( ), for Example 2. The corresponding analytical solu-

tions (7.30) are shown by continuous line (—–) in (c) and (d).. . . . 170

List of Tables xxvi

7.10 (a) The objective functionFλ, (b) the RMSEs forr(t) (− − −) and

s(x) (· · · ) obtained using the hybrid-order regularisation (7.26) with

regularisation parameterλopt = 2 for noise levelǫ0 = 0.1, and the

numerical results(− · −) for (c) r(t) and (d)s(x) obtained using the

minimisation process after 17 (unfixed) iterations, for Example 2. The

corresponding analytical solutions (7.30) are shown by continuous line

(—–) in (c) and (d). . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.11 The numerical results for (a)χ(t) and (b)β(x) obtained using the BEM

for the direct problem withN = N0 ∈ 10(− · −), 20(· · · ), 40(−−−), for Example 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.12 (a) The objective functionFλ and the numerical results for (b)r(t) and

(c) s(x) obtained with the hybrid-order regularisation (7.26) withreg-

ularisation parameterλ = 2×10−4 for p ∈ 1(−·−), 5(· · · ), 10(−−−)% noisy data for Example 3. The corresponding analytical solu-

tions (7.33) and (7.34) are shown by continuous line (—–) in (b) and

(c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

List of Tables

2.1 The condition numbers of the matrixX in equation (2.48) forN =

N0 ∈ 20, 40, 80 for Example 1 Cases 1–6. . . . . . . . . . . . . . . 34

2.2 The RMSE for the zeroth- and first-order Tikhonov regularisation for

p ∈ 0, 1, 3% noise, for Example 1 Case 1. . . . . . . . . . . . . . . 34










p ∈ 0, 1, 3, 5% noise, for Example 1 Case 6. . . . . . . . . . . . . . 48

2.8 The RMSE for the first-order Tikhonov regularisation forp ∈ 0, 0.1, 0.5, 1%noise, for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1 The RMSE for the TSVD, ZOTR, FOTR, SOTR forp ∈ 0, 1, 3, 5%noise, for Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.2 The RMSE for the ZOTR, FOTR, SOTR forp ∈ 1, 3, 5% noise and

N = 40,N0 = 30, for Example 2. . . . . . . . . . . . . . . . . . . . 79

xxvii

List of Tables xxviii

5.1 The RMSE foru(1, t), ux(0, t), ux(1, t) andE(t) obtained using the

BEM for the direct problem withN = N0 ∈ 20, 40, 80, for Example 1.107

5.2 The regularisation parametersλ and the RMSE forr(t), u(1, t),ux(0, t)

andux(1, t), obtained using the BEM withN = N0 = 40 combined

with the second-order Tikhonov regularisation forp ∈ 0, 1, 3, 5%noise, for Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.3 The regularisation parametersλ and the RMSE forr(t), u(1, t),ux(0, t)

andux(1, t), obtained using the BEM withN = 40 andN0 = 30 com-

bined with the second-order Tikhonov regularisation forp ∈ 0, 1, 3, 5%noise, for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.4 The regularisation parametersλ given by the discrepancy principle and

by the GCV, and the RMSE forr(t), u(1, t), ux(0, t) andux(1, t), ob-

tained using the BEM withN = 40 andN0 = 30 combined with the

second-order Tikhonov regularisation forp ∈ 0, 1, 3, 5% noise. The

optimal regularisation parameter given by the minimum of RMSE of

r(t) is also included, for Example 3. . . . . . . . . . . . . . . . . . . 120

6.1 The condition numbers of the matrixX in equation (6.21) for various

N = N0 ∈ 20, 40, 80 andX0 ∈ 14, 12, 34, for Example 1. . . . . . . 133

6.2 The RMSE forr(t), s(x), ux(0, t), andux(1, t) obtained using the

SVD, TSVD, ZOTR, FOTR, and SOTR, forp ∈ 0, 1%, for Example 1.142

6.3 The RMSE forr(t), s(x), ux(0, t), andux(1, t) obtained using the

SVD, TSVD, ZOTR, FOTR, and SOTR, forp ∈ 0, 1%, for Example 2.144

7.1 The RMSE foru(0, t), u(0.1, t), χ(t) andβ(x), obtained using the

BEM for the direct problem withN = N0 ∈ 10, 20, 40, for Example 1.158

7.2 The RMSE forr(t), s(x), u(0, t), u(0.1, t) for exact data, Example 1. 165

7.3 The RMSE forr(t) ands(x), for the noise levelsǫ0 ∈ 0, 0.01, 0.1,

for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Chapter 1

General Introduction

1.1 Introduction

Inverse problems are becoming an essential part in the development of several appli-

cations in science and engineering such as in medical diagnosis and therapy, ground-

water/air pollution phenomena, or the designing of thermalequipment, systems and

instruments. Such problems, particularly for the heat equation, have important appli-

cations in the field of applied sciences such as in melting andfreezing processes, the

designing and manufacturing areas in which the strength of heat sources is not ex-

actly recognised, especially in the discovery of the quantity of energy generation in a

computer chip, in a microwave heating process, or in a chemical reaction process.

In this thesis, the interest is specialised to solve severalinverse source problems for

the heat equation using the boundary element method (BEM).

1.2 Inverse and ill-posed problems

A direct problem consists of solving a system where an input cause is given and an

output effect is desired. However, if the situation is reversed then we have an inverse

problem which is in general ill-posed (improperly-posed, incorrectly-posed). For more

definitions and examples of inverse and ill-posed problems see the excellent review by

1

Chapter 1. 2

Kabanikhin [31].

The study of ill-posed problems began in the early 20th century through the defi-

nition of well-posedness given by J. Hadamard in 1902. In thesense of Hadamard, a

mathematical problem is well-posed if it satisfies the following properties:

• Existence: For all (suitable) data, there exists a solution of the problem (in an

appropriate sense).

• Uniqueness: For all (suitable) data, the solution is unique.

• Stability: The solution depends continuously on its data (i.e. small perturbations

in the input data do not result in large perturbations in the solution).

According to above definition, any mathematical problem is ill-posed if any one of

these three conditions is violated. In the cases investigated in this thesis, the problems

violate the third condition, i.e. stability.

The main purpose of this thesis focuses on applying BEM to inverse heat source

problems, which are in generally ill-posed in the sense thatsmall measurement errors

greatly magnify the sought solutions.

1.3 The boundary element method (BEM)

One of the main advantage of the BEM over domain discretisation methods such as

the finite-difference method (FDM) or the finite element mehtod (FEM) is that the

discretisation is necessary only on the boundary, i.e. the BEM uses less number of

nodes and elements when compared to the FDM and the FEM. The main idea of the

BEM, which is based on using the Green’s identity and the fundamental solution, is

to find the solution inside the domain by using the solution tothe partial differential

equation (PDE) on the boundary only.

The mathematical background of the BEM is represented by theknowledge of the

fundamental solution and the application of the Green’s identities. We first introduce

Chapter 1. 3

the Heaviside step function and the Dirac delta distribution as follows:

The Heaviside step function:

H(t) =

1, if t > 0,

0, if t ≤ 0.

The Dirac delta distribution function:

δ(x, ξ) = δ(x− ξ) =

0, if x 6= ξ,

∞, if x = ξ.

The fundamental properties of the Dirac delta distributionare

δ(x) = H ′(x),

∫

Ω

f(ξ)δ(x, ξ) dξ = f(x), x ∈ Ω.

Basically, the one-dimensional transient heat equation isgoverned by the partial

differential heat operatorL := ∂2

∂x2− ∂

∂t. Let L > 0 andT > 0 be the length of the

space domain and the time duration, respectively, and definethe solution domain

DT := (0, L)× (0, T ]. (1.1)

Consider the classical heat equation

Lu(x, t) = ∂2u

∂x2(x, t)− ∂u

∂t(x, t) = 0, (x, t) ∈ DT . (1.2)

A functionG(x, t, y, τ) is called a fundamental solution for the heat equation (1.2)if

L∗G(x, t, y, τ) = −δ(x, t; y, τ) = −δ(|x− y|, |t− τ |), (1.3)

whereL∗ = ∂2

∂x2+ ∂

∂tis the adjoint ofL, (x, t) is a field point, and(y, τ) is a source

point. Solving (1.3) using the method of Fourier transform gives the fundamental

Chapter 1. 4

solution, see [60],

G(x, t, y, τ) =H(t− τ)√

4π(t− τ)exp

(

−(x− y)2

4(t− τ)

)

. (1.4)

In order to develop the BEM, let us introduce the Green’s identities, as follows:

∫

Ω

(

U∇2V − V∇2U)

dΩ =

∫

∂Ω

(

U∂V

∂n− V

∂U

∂n

)

dS,

∫

Ω

(

U∇2V +∇U · ∇V)

dΩ =

∫

∂Ω

U∂V

∂ndS,

(1.5)

for any functionsU, V ∈ C2(Ω), wheren is the outward normal to the boundary∂Ω

of the bounded domainΩ.

1.4 The BEM for solving one-dimensional direct heat

problem

In order to understand how the BEM performs, let us consider the direct classical (one-

dimensional) heat conduction problem which requires finding the temperatureu(x, t)

satisfying the heat equation

ut = uxx + F (x, t), (x, t) ∈ DT , (1.6)

whereF is a heat source, subject to the initial condition

u(x, 0) = u0(x), x ∈ [0, L], (1.7)

and the Neumann boundary conditions

ux(0, t) = µ1(t), ux(L, t) = µ2(t), t ∈ (0, T ], (1.8)

Chapter 1. 5

(or the Dirichlet boundary conditions)

u(0, t) = µ1(t), u(L, t) = µ2(t), t ∈ (0, T ]. (1.9)

Mixed or Robin boundary conditions can also be considered. For using the BEM, we

first multiply (1.6) byG and integrate overDT to result in

∫

DT

G(x, t, y, τ)∂u

∂τ(y, τ) dydτ =

∫

DT

G(x, t, y, τ)∂2u

∂y2(y, τ) dydτ

+

∫

DT

G(x, t, y, τ)F (y, τ) dydτ.

Using the Green’s identities (1.5) gives

∫

DT

G(x, t, y, τ)∂u

∂τ(y, τ) dydτ

=

∫ T

0

[

G(x, t, ξ, τ)∂u

∂n(ξ)(ξ, τ)− u(ξ, τ)

∂G

∂n(ξ)(x, t, ξ, τ)

]

ξ∈0,L

dτ

+

∫

DT

u(y, τ)∂2G

∂y2(x, t, y, τ) dydτ +

∫

DT

G(x, t, y, τ)F (y, τ) dydτ, (1.10)

wheren is the outward unit normal to the space boundary0, L, i.e. ∂∂n(ξ)

= − ∂∂ξ

for

ξ = 0, and ∂∂n(ξ)

= ∂∂ξ

for ξ = L. Then, using that the fundamental solution satisfies

(1.3) and the property of the Dirac delta function result in the integral equation

η(x)u(x, t) =

∫ t

0

[

G(x, t, ξ, τ)∂u

∂n(ξ)(ξ, τ)− u(ξ, τ)

∂G

∂n(ξ)(x, t, ξ, τ)

]

ξ∈0,L

dτ

+

∫ L

0

G(x, t, y, 0)u(y, 0) dy+

∫ L

0

∫ T

0

G(x, t, y, τ)F (y, τ) dτdy,

(x, t) ∈ [0, L]× (0, T ], (1.11)

whereη(0) = η(L) = 12

andη(x) = 1 for x ∈ (0, L).

The discretisation of the integral equation (1.11) is performed by dividing the

boundaries0 × (0, T ] andL × (0, T ] into a series ofN small boundary elements

Chapter 1. 6

[tj−1, tj] for j = 1, N , tj =jTN, j = 0, N , whilst the space domain[0, L]× 0 is dis-

cretised into a series ofN0 small cells[xk−1, xk] for k = 1, N0, xk = kN0

, k = 0, N0.

Over each boundary element(tj−1, tj], the temperatureu and the flux∂u∂n

are assumed

to be constant and take their values at the midpointtj =tj−1 + tj

2, i.e.

u(0, t) = u(0, tj) =: h0j , u(L, t) = u(L, tj) =: hLj , t ∈ (tj−1, tj] (1.12)

∂u

∂n(0, t) =

∂u

∂n(0, tj) =: q0j ,

∂u

∂n(L, t) =

∂u

∂n(L, tj) =: qLj , t ∈ (tj−1, tj ]. (1.13)

Note that sincen is the outward unit normal to the (one-dimensional) space boundary,

then

q0j = −∂u∂x

(0, tj), qLj =∂u

∂x(L, tj). (1.14)

In each cell[xk−1, xk], the temperatureu is assumed to be constant and takes its value

at the midpointxk =xk−1 + xk

2, i.e.

u(x, 0) = u(xk, 0) =: u0,k, x ∈ (xk−1, xk]. (1.15)

Also, for the source functionF (x, t), we assume the piecewise constant approximation

in time as

F (x, t) = F (x, tj), t ∈ (tj−1, tj ]. (1.16)

With these approximations, the integral equation (1.11) isdiscretised as

η(x)u(x, t) =

N∑

j=1

[A0j(x, t)q0j + ALj(x, t)qLj −B0j(x, t)h0j −BLj(x, t)hLj ]

+

N0∑

k=1

Ck(x, t)u0,k +N∑

j=1

D0,j(x, t), (x, t) ∈ [0, L]× (0, T ], (1.17)

where the integral coefficients are given by

Aξj(x, t) =

∫ tj

tj−1

G(x, t, ξ, τ)dτ, ξ = 0, L, (1.18)

Chapter 1. 7

Bξj(x, t) =

∫ tj

tj−1

∂G

∂n(ξ)(x, t, ξ, τ)dτ, ξ = 0, L, (1.19)

Ck(x, t) =

∫ xk

xk−1

G(x, t, y, 0)dy, (1.20)

and the double integral source term is given by

D0,j(x, t) =

∫ tj

tj−1

∫ L

0

G(x, t, y, τ)F (y, tj) dydτ. (1.21)

The integrals in expressions (1.18)–(1.20) can be evaluated analytically as, see [15],

Aξj(x, t) =

0 ; t ≤ tj−1,√

t− tj−1

π; tj−1 < t ≤ tj , x = ξ,

|x− ξ|2√π

(

e−z2

0

z0−√

πerfc(z0)

)

; tj−1 < t ≤ tj , x 6= ξ,

√

t− tj−1

π−√

t− tjπ

; t > tj, x = ξ,

|x− ξ|2√π

(

e−z2

0

z0− e−z

2

1

z1+√π (erf(z0)− erf(z1))

)

; t > tj, x 6= ξ,

(1.22)

Bξj(x, t) =

0 ; t ≤ tj−1,

0 ; tj−1 < t ≤ tj, x = ξ,

−erfc(z0)2

; tj−1 < t ≤ tj, x 6= ξ,

erf(z0)− erf(z1)2

; t > tj ,

(1.23)

Ck(x, t) =1

2

[

erf

(

x− xk−1

2√t

)

− erf

(

x− xk

2√t

)]

, (1.24)

whereξ ∈ 0, L, z0 =|x− ξ|

2√t− tj−1

, z1 =|x− ξ|2√t− tj

and erf, erfc are the error func-

tions defined by erf(x) =2√π

∫ x

0

e−σ2

dσ, erfc(x) = 1 − erf(x), respectively. Mean-

Chapter 1. 8

while the double integral (1.21) becomes

D0,j(x, t) =

∫ tj

tj−1

∫ L

0

G(x, t, y, τ)F (y, tj) dydτ =

∫ L

0

F (y, tj)Ayj(x, t) dy,

and can be evaluated using the midpoint rule for numerical integration.

Hence on considering the BEM, we apply the initial condition(1.7) at the nodes

xk for k = 1, N0, as in (1.15), and the integral equation (1.17) at the boundary nodes

(0, ti) and(L, ti) for i = L,N . This gives the system of2N linear equations

Aq¯− Bh

¯+ Cu

¯0+ d

¯= 0

¯, (1.25)

where

A =

A0j(0, ti) ALj(0, ti)

A0j(L, ti) ALj(L, ti)

2N×2N

,

B =

B0j(0, ti) +12δij BLj(0, ti)

B0j(L, ti) BLj(L, ti) +12δij

2N×2N

, C =

Ck(0, ti)

Ck(L, ti)

2N×N0

,

q¯=

q0j

qLj

2N

, h¯=

h0j

hLj

2N

, u¯0

=[

u0,k

]

N0

, d¯=

∑Nj=1D0,j(0, ti)

∑Nj=1D0,j(L, ti)

2N

,

whereδij is the Kronecker delta symbol, defined byδij = 1 for i = j, andδij = 0

for i 6= j. Note that matrix termB also includes the contribution from the left-hand

side of equation (1.17). At this stage, we can find the boundary temperature h¯, if the

Neumann boundary conditions (1.8) are prescribed as

h¯= B−1

(

Aq¯+ Cu

¯0+ d

¯

)

.

Whereas if the Dirichlet boundary conditions (1.9) are prescribed, we can then obtain

the heat flux q¯

as

q¯= A−1 (Bh

¯− Cu

¯0− d

¯) .

Chapter 1. 9

1.5 Condition number

The insight into the degree of conditioning of the system of equations (1.25) is merely

given by the condition number of a matrix herein defined as theratio between the

largest to the smallest singular values. Obviously, the larger the condition number is

the more ill-conditioned is our system of equations.

1.6 Regularisations

Inverse problems are well-known to be in general ill-posed by violating the stability

condition at least. Upon discretisation, this results in anill-conditioned systems of

equations to be solved. To deal with these difficulties the inverse problem is usually

solved as an optimisation problem with regularisation in order to achieve the stability

of the solution. Below we briefly describe two such classicalmethods of regularisation.

1.6.1 The truncated singular value decomposition (TSVD)

Suppose we wish to solve the system ofM linear equations withN unknowns

Xr¯= y

¯ǫ, (1.26)

whereyǫ is a noisy perturbation of the exact right-hand side vector y¯, i.e. ‖y−yǫ‖ ≈ ǫ.

We first decompose the matrixX in the form,

X = UΣV T, (1.27)

whereU = [U¯1,U¯2, . . . ,U¯N

] andV = [V¯1,V¯2, . . . ,V¯N

] areM × N matrices with

columns U¯j

and V¯j

for j = 1, N , such thatUTU = I = V TV , and

Σ = diag(σ1, σ2, . . . , σN) is anN diagonal matrix containing the singular values of

Chapter 1. 10

the matrixX, σj for j = 1, N , in decreasing order

σ1 ≥ σ2 ≥ · · · ≥ σN ≥ 0.

Then the matrix system (1.26) can be reformed to obtain the singular value decompo-

sition (SVD) solution as follows:

r¯=

(

N∑

j=1

1

σjV¯j

· U¯

Tj

)

y¯ǫ. (1.28)

In MATLAB, this decomposition is operated using the command[U,Σ, V ] = svd(X)

or [U,Σ, V ] = svds(X,N). For ill-posed problems, the truncation of (1.28) is needed

to be considered as a regularisation method, by omitting itslastN −Nt small singular

values, whereNt denotes the truncation level. This way, the regularised solution is

given by

r¯Nt

=

(

Nt∑

j=1

1

σjV¯j

· U¯

Tj

)

y¯ǫ, (1.29)

which is simply a truncated SVD (TSVD) stable solution of thefull SVD unstable so-

lution (1.28). And the MATLAB command for the TSVD becomes[UNt,ΣNt

, VNt] =

svds(X,Nt) whereΣNt= diag(σ1, σ2, . . . , σNt

) andUNt=[

U¯1,U¯2, . . . ,U¯NT

]

, VNt=

[

V¯1,V¯2, . . . ,V¯Nt

]

.

1.6.2 The Tikhonov regularisation

Alternatively, the Tikhonov regularisation is another wayof obtaining a stable solution

of the ill-conditioned system of equations (1.26). This method is based on minimising

the regularised linear least-squares objective function,[42, 57],

‖Xr¯− y

¯ǫ‖2 + λ‖Rr‖2 (1.30)

Chapter 1. 11

whereR is a (differential) regularisation matrix of orderk ∈ 0, 1, 2, . . . imposing a

Ck-smoothing constraint on the solution, andλ > 0 is a regularisation parameter to be

prescribed. Note that the norm‖ · ‖ is defined as the Euclidean norm of vector. In this

study, we are considering the order of regularisation matrix R to be order zero, one,

two as defined by [15, 57],

R0 =

1 0 0 .

0 1 0 .

0 0 1 .

. . . .

, the zeroth-order regularisation, (1.31)

R1 =

1 −1 0 0 .

0 1 −1 0 .

0 0 1 −1 .

. . . . .

, the first-order regularisation, (1.32)

R2 =

1 −2 1 0 0 .

0 1 −2 1 0 .

0 0 1 −2 1 .

. . . . . .

, the second-order regularisation. (1.33)

On solving the minimisation of (1.30) one obtains the regularised solution

r¯λ

=(

XTX + λRTR)−1

XTy¯ǫ. (1.34)

1.6.3 Choice of the regularisation parameter

The regularisation parameterλ is very important in (1.34) (also the truncation levelNt

in (1.29)) and it can be chosen according to many criteria, e.g. theL-curve method

[16], the generalised cross-validation (GCV) [63], or the discrepancy principle [40].

TheL-curve method suggests choosingλ at the corner of theL-curve which is a plot

of the norm of the residual‖Xr¯λ−y

¯ǫ‖ versus the solution norm‖r

¯λ‖. Alternatively, the

Chapter 1. 12

discrepancy principle choosesλ > 0 such that the residual‖Xr¯λ

− y¯ǫ‖ ≈ ǫ. Whereas

the GCV criterion suggests choosing the parameterλ as the minimum of the GCV

function,

GCV (λ) =‖Xr

¯λ− y

¯ǫ‖2

[trace(I −X(XTX + λRTR)−1XT)]2, λ > 0. (1.35)

Note that both theL-curve and the GCV criteria are heuristic methods, which arenot

always convert [58], because they do not require the knowledge of the level of noiseǫ.

Then these two methods do not guarantee to give the regularisation parameter.

1.7 Purpose of the thesis

In this thesis, we mainly consider inverse heat source problems for the heat equation

(1.6), whereu is the unknown temperature andF is a heat source term to be identified.

We focus on the identification of the source termF (x, t) in (1.6) in various special

cases. This approach is necessary because otherwise there will be no unique solution

to the inverse problem unlessu(x, t) is specified or measured throughout the whole

solution domainDT , [56].

Moreover, even though uniqueness of solution can be ensuredby restricting the

source term to be of certain special forms, e.g. space-dependent, time-dependent, ad-

ditive or multiplicative, the inverse problem is still ill-posed in the sense that the con-

tinuous dependence upon the input data is violated (small errors in the input data give

rise to large errors in the estimated results). This has to bedealt with by using some

sort of regularisation, e.g. the TSVD as described in Subsection 1.6.1, the Tikhoknov

regularisation as described in Subsection 1.6.2 [1, 15, 63], the iterative algorithm [30],

the variational method [29], the augmented Tikhonov regularisation derived from a

Bayesian perspective [65], the mollification methods [68, 69], the smoothing spline

approximation [59], etc.

The structure of the thesis is as follows. In Chapter 1, the background knowledge

Chapter 1. 13

of inverse and ill-posed problems is provided. The BEM is detailed together with the

application to the classical heat equation.

In Chapter 2, three general boundary conditions of inverse heat source problems are

considered to determine the time-dependent heat sourcer(t) in F (x, t) = r(t)f0(x, t)+

f1(x, t) and the temperatureu(x, t) in the heat equation (1.6), subject to the initial con-

dition (1.7), and the following three general nonlocal boundary and overdetermination

conditions:

γ11(t)u(0, t) + γ12(t)u(L, t) + γ13(t)ux(0, t) + γ14(t)ux(L, t) = k1(t)



, (1.36)

where(ki)i=1,3 are given functions and(γij)i=1,3,j=1,4 is a given matrix of coefficients

having rank 3. The BEM is combined with the Tikhonov regularisation in order to

obtain an accurate and stable numerical solution.

In Chapter 3, we investigate an identification of the time-dependent heat source, i.e.

we seekr(t) in F (x, t) = r(t)f(x, t), together with the temperatureu(x, t) in the heat

equation (1.6), subject to the initial condition (1.7), theperiodic and Robin boundary

conditions

u(0, t) = u(1, t), t ∈ [0, T ], (1.37)

ux(0, t) + αu(0, t) = 0, t ∈ [0, T ], (1.38)

whereα 6= 0 is a given constant, and the integral additional measurement

∫ 1

0

u(x, t)dx = E(t), t ∈ [0, T ]. (1.39)

In this inverse problem, the BEM is developed as a numerical method and combined

with two case studies of the regularisation method. Firstly, we apply the BEM together

with the TSVD method in order to obtain a stable solution, andthen the BEM is con-

Chapter 1. 14

sidered again and combined with various orders of Tikhonov’s regularisation method.

In Chapter 4, we determine the time-dependent blood perfusion coefficient function

P (t) ≥ 0 and the temperatureu(x, t) in the following bioheat equation

ut(x, t) = uxx(x, t)− P (t)u(x, t) + f(x, t), (x, t) ∈ (0, 1)× (0, T ], (1.40)

wheref is a given heat source term. We subject this bioheat equationto the ini-

tial condition (1.7), the boundary conditions (1.37) and (1.38), and the integral over-

determination condition (1.39). A simple transformation is used to reduce the bioheat

equation (1.40) to the classical heat equation (1.6). The BEM for the heat equation

is employed, together with either the second-order Tikhonov regularisation combined

with finite differences, or with a smoothing spline regularisation technique for com-

puting the first-order derivative of a noisy function.

Chapter 5 presents an investigation for the identification of the time-dependent heat

sourcer(t) in F (x, t) = r(t)f(x, t) and the temperatureu(x, t) in the heat equation in

(1.6), subject to the initial condition (1.7), the non-classical boundary condition

auxx(1, t) + αux(1, t) + bu(1, t) = 0, t ∈ [0, T ], (1.41)

wherea, b, α are given numbers not simultaneously equal to zero, and the over-

determination condition (1.39). We are using the same techniques as before.

More challenging, the purpose in Chapter 6 is the simultaneous determination of

an additive space- and time-dependent heat sources, i.e. identifying the unknown com-

ponentsr(t) ands(x) in the source termF (x, t) = r(t)f(x, t) + s(x)g(x, t) + h(x, t),

together with the temperatureu(x, t) in the heat equation (1.6), subject to the initial

condition (1.7), the Dirichlet boundary conditions,

u(0, t) = µ0(t), u(L, t) = µL(t), t ∈ [0, T ], (1.42)

and additional conditions. These latter ones consist of a specified temperature measure-

Chapter 1. 15

ment at an internal pointX0 ∈ (0, L), a time-average temperature, and an additional

fixing conditions, as follows:

u (X0, t) = χ(t), t ∈ [0, T ], (1.43)∫ T

0

u(x, t) dt = ψ(x), x ∈ [0, L], (1.44)

s(X0) = S0. (1.45)

The mathematical problem is linear but ill-posed since the continuous dependence on

the input data is violated. In discretised form the problem reduces to solving an ill-

conditioned system of linear equations. We investigate theperformances of several

regularisation methods, i.e. the TSVD and the Tikhonov regularisation, and examine

their stability with respect to noise in the input data.

A nonlinear heat source problem is finally studied in Chapter7. This consists

of the simultaneous determination of multiplicative space- and time-dependent source

componentsf(t) andg(x) in F (x, t) = r(t)s(x), in the heat equation (1.6), subject to

the initial condition (1.7), the homogeneous Neumann boundary conditions

ux(0, t) = ux(L, t) = 0, t ∈ [0, T ], (1.46)

the specified interior measurement (1.43), the final time temperature measurement at

the ‘upper-base’ final timet = T , and an additional fixing condition, as follows:

u(x, T ) = β(x), x ∈ [0, L], (1.47)

s(X0) = S0. (1.48)

For the numerical discretisation, the BEM combined with a regularised nonlinear op-

timisation are utilised.

Finally, in Chapter 8, the conclusions which summarise the main work of this thesis

and possible future work are highlighted.

Chapter 2. 16

Chapter 2

Determination of a Time-dependent

Heat Source from Nonlocal Boundary

Conditions

2.1 Introduction

Recently, nonlocal boundary and overdetermination conditions have become a centre

of interest in the mathematical formulation and numerical solution of several inverse

and improperly posed problems in transient heat conduction, see e.g. [23, 24, 33, 51],

to mention only a few. They opened a new area of applied numerical and mathematical

modelling research. Practical applications of nonlocal boundary value problems are

encountered in chemical diffusion for heat conduction in biological processes, see e.g.

[11, 41, 46]. For example, in multiphase flows involving fluids, solids and gases, the

heat flux is often taken to be proportional to the difference in boundary temperature

between the various phases, and the quantitiesγij , i = 1, 3, j = 1, 4, present in the

nonlocal boundary condition (2.3) below (see also equation(1.36)) represent those

proportionality factors.

In this chapter, we consider obtaining the numerical solution of several inverse

17

Chapter 2. 18

time-dependent heat source problems for the heat equation with non-local boundary

and overdetermination conditions whose unique solvabilities have previously been in-

vestigated/established by Ivanchov [28]. The mathematical inverse formulations are

described in Section 2.2. Since the inverse problems under investigations are linear,

but ill-posed (in the sense that the continuous dependence upon the input data is vio-

lated), the numerical method is based on the boundary element direct solver combined

with the Tikhonov regularisation, as described in Section 2.3. The choice of the regu-

larisation parameter in the latter procedure is based on thediscrepancy principle, [40].

The above combination yields accurate and stable numericalsolutions, as it will be

presented and discussed in Section 2.4. Finally, Section 2.5 highlights the conclusions

of this chapter.

2.2 Mathematical formulation

Consider the problem of finding the time-dependent heat sourcer(t) ∈ C([0, T ]) and

the temperatureu(x, t) ∈ C2,1(DT ) ∩ C1,0(DT ) which satisfy the heat conduction

equation

ut = uxx + r(t)f(x, t) + h(x, t), (x, t) ∈ DT , (2.1)

subject to the initial condition (1.7), namely

u(x, 0) = u0(x), x ∈ [0, L], (2.2)

and the following general boundary and overdetermination conditions:




(2.3)

Chapter 2. 19

where

f, h ∈ C1,0(DT ), u0 ∈ C2([0, L]), ki ∈ C1([0, T ]), i = 1, 3, (2.4)

and the matrixγ = (γij)i=1,3,j=1,4 ∈ C1([0, T ]) has rank 3 for allt ∈ [0, T ].

Actually, this inverse problem was studied theoretically by Ivanchov [28] who es-

tablished its unique solvability. Moreover, by assuming, without any loss of generality,

that the same third-order minor of the matrixγ is non-zero we can express three of the

four boundary datau(0, t), u(L, t), ux(0, t), andux(L, t) in terms of the fourth one and

distinguish the following six cases:

Case 1 ux(0, t) = µ1(t), ux(L, t) = µ2(t), (2.5)

v1(t)u(0, t) + v2(t)u(L, t) = k(t); (2.6)

Case 2 u(0, t) = µ1(t), ux(L, t) = µ2(t), (2.7)

v1(t)ux(0, t) + v2(t)u(L, t) = k(t); (2.8)

Case 3 u(0, t) = µ1(t), u(L, t) = µ2(t), (2.9)

v1(t)ux(0, t) + v2(t)ux(L, t) = k(t); (2.10)

Case 4 u(0, t) = µ1(t), ux(L, t) + v1(t)u(L, t) = µ2(t), (2.11)

ux(0, t) + v2(t)ux(L, t) = k(t); (2.12)

Case 5 ux(0, t) = µ1(t), ux(L, t) + v1(t)u(L, t) = µ2(t), (2.13)

u(0, t) + v2(t)u(L, t) = k(t); (2.14)

Case 6 ux(0, t)− v1(t)u(0, t) = µ1(t), ux(L, t) + v2(t)u(L, t) = µ2(t), (2.15)

v3(t)u(0, t) + v4(t)u(L, t) = k(t), (2.16)

for t ∈ [0, T ], wherek ∈ C1([0, T ]) is a given function resulted from manipulating the

system (2.3). The other mixed boundary conditions cases corresponding to Cases 2,

4–6 can be reduced to these ones by the change of variabley = L− x.

The following Theorems 2.2.1–2.2.5 from [28] give the unique solvability, i.e. ex-

Chapter 2. 20

istence and uniqueness of the solutions of the inverse problem (2.1)–(2.3) in all the

above six cases.

Theorem 2.2.1 Assume that the regularity conditions(2.4)are satisfied and that:

(i) v1, v2 ∈ C1([0, T ]), v21(t) + v22(t) > 0, t ∈ [0, T ];

(ii) v1(t)f(0, t) + v2(t)f(L, t) 6= 0, t ∈ [0, T ];

(iii) µ1(0) = u′0(0), µ2(0) = u′0(L), v1(0)u0(0) + v2(0)u0(L) = k(0).

Then the inverse problem(2.1), (2.2), (2.5), (2.6)representing Case 1 is uniquely solv-

able.

Theorem 2.2.2 Assume that, in addition to conditions(2.4)and(i) of Theorem 2.2.1,

the following conditions are satisfied:

(i) v1(t)f(0, t) 6= 0, t ∈ [0, T ];

(ii) µ1(0) = u0(0), µ2(0) = u′0(L), v1(0)u′0(0) + v2(0)u0(L) = k(0).

Then the inverse problem(2.1), (2.2), (2.7), (2.8)representing Case 2 is uniquely solv-

able.

Theorem 2.2.3 Assume that, in addition to conditions(2.4), (i) and (ii) of Theorem

2.2.1, the following conditions are satisfied:

µ1(0) = u0(0), µ2(0) = u0(L), v1(0)u′0(0) + v2(0)u

′0(L) = k(0).

Then the inverse problem(2.1), (2.2), (2.9), (2.10) representing Case 3 is uniquely

solvable.


(i) v1 ∈ C[0, T ], v2, µi ∈ C1[0, T ], i = 1, 2, v1(t) > 0, t ∈ [0, T ];

Chapter 2. 21

(ii) in Case 4,

f(0, t)− v2(t)f(L, t) 6= 0, t ∈ [0, T ],

µ1(0) = u0(0), µ2(0) = u′0(L) + v1(0)u0(L), k(0) = u′0(0) + v2(0)u′0(L);

(iii) in Case 5,

f(0, t) + v2(t)f(L, t) 6= 0, t ∈ [0, T ],

µ1(0) = u′0(0), µ2(0) = u′0(L) + v1(0)u0(L), k(0) = u0(0) + v2(0)u0(L).

Then the inverse problem(2.1), (2.2), (2.11), (2.12) representing Case 4, and(2.1),

(2.2), (2.13), (2.14)representing Case 5 are uniquely solvable.


(i) vi ∈ C[0, T ], v3, v4, µi ∈ C1[0, T ], vi(t) > 0, i = 1, 2, t ∈ [0, T ];

(ii) v1(t)f(0, t) + v2(t)f(L, t) 6= 0, v23(t) + v24(t) > 0, t ∈ [0, T ];

(iii) µ1(0) = u′0(0)− v1(0)u0(0), µ2(0) = u′0(L) + v2(0)u0(L),

k(0) = v3(0)u0(0) + v4(0)u0(L).

Then the inverse problem(2.1), (2.2), (2.15), (2.16) representing Case 6 is uniquely

solvable.

Although the problems of Cases 1–6 are uniquely solvable, they are still ill-posed

since small errors in the input datak(t) lead to large errors in the output source solution

r(t). In the next subsection we describe how the BEM discretisingnumerically the heat

equation (2.1) can be used in conjunction with the Tikhonov regularisation in order to

obtain a stable solution.

Chapter 2. 22


In the numerical process, we employ the BEM as introduced in Section 1.3. For the

heat equation (2.1) we then obtain the integral equation

η(x)u(x, t) =

∫ t

0

[

G(x, t, ξ, τ)∂u

∂n(ξ)(ξ, τ)− u(ξ, τ)

∂G

∂n(ξ)(x, t, ξ, τ)

]

ξ∈0,L

dτ

+

∫ L

0

G(x, t, y, 0)u(y, 0) dy+

∫ L

0

∫ t

0

G(x, t, y, τ)r(τ)f(y, τ) dτdy

+

∫ L

0

∫ t

0

G(x, t, y, τ)h(y, τ) dτdy, (x, t) ∈ [0, L]× (0, T ). (2.17)

We use the constant BEM (CBEM) with the midpoint approximations (1.12), (1.13)

and (1.15). Nevertheless, higher-order, e.g. linear boundary element approximations

will be more accurate than constant boundary elements. Thisimprovement in accuracy

will be significant in higher-dimension, see e.g. [49], but in our one-dimensional time-

dependent setting the use of the CBEM approximation was found sufficiently accurate.

With this, the integral equation (2.17) can be approximatedas

η(x)u(x, t) =

N∑

j=1

[A0j(x, t)q0j + ALj(x, t)qLj − B0j(x, t)h0j − BLj(x, t)hLj]

+

N0∑

k=1

Ck(x, t)u0,k + d(x, t) + d0(x, t), (2.18)

where the coefficientsAξj, Bξj, ξ ∈ 0, 1, andCk are given by (1.18)–(1.20) and can

be evaluated analytically as in (1.22)–(1.24), respectively. Whereas the double integral

source termsd andd0 are given by

d(x, t) =

∫ L

0

∫ t

0

G(x, t, y, τ)r(τ)f(y, τ) dτdy, (2.19)

d0(x, t) =

∫ L

0

∫ t

0

G(x, t, y, τ)h(y, τ) dτdy, (2.20)

Chapter 2. 23

and can be evaluated by assuming the piecewise constant approximations for the source

functionsf(x, t), h(x, t), andr(t), i.e.

f(x, t) = f(x, tj), h(x, t) = h(x, tj), r(t) = r(tj) =: rj, (2.21)

for t ∈ (tj−1, tj] and j = 1, N . By these approximations, the integrals (2.19) and

(2.20) become

d(x, t) =

∫ t

0

r(τ)

∫ L

0

G(x, t, y, τ)f(y, τ) dydτ =

N∑

j=1

Dj(x, t)rj, (2.22)

d0(x, t) =

∫ t

0

∫ L

0

G(x, t, y, τ)h(y, τ) dydτ =N∑

j=1

D0,j(x, t), (2.23)

where

Dj(x, t) =

∫ tj

tj−1

∫ L

0

G(x, t, y, τ)f(y, tj) dydτ =

∫ L

0

f(y, tj)

∫ tj

tj−1

G(x, t, y, τ) dτdy

=

∫ L

0

f(y, tj)Ayj(x, t) dy,

D0,j(x, t) =

∫ tj

tj−1

∫ L

0

G(x, t, y, τ)h(y, tj) dydτ =

∫ L

0

h(y, tj)

∫ tj

tj−1

G(x, t, y, τ) dτdy

=

∫ L

0

h(y, tj)Ayj(x, t) dy,

are evaluated numerically using the midpoint integral approximation. Here, the integral

equation (2.18) can be rewritten as

η(x)u(x, t) =

N∑

j=1


+

N0∑

k=1

Ck(x, t)u0,k +

N∑

j=1

Dj(x, t)rj +

N∑

j=1

D0,j(x, t). (2.24)

Chapter 2. 24

Applying (2.24) at the boundary nodes(0, ti) and(L, ti) for i = 1, N , we obtain the

following system of2N equations

Aq¯−Bh

¯+ Cu

¯0+Dr

¯+ d

¯= 0

¯, (2.25)

where matricesA, B, C, D and vectors q¯, h

¯, u

¯0, r¯, d

¯are defined as same as in Section

1.4, andD =

Dj(0, ti)

Dj(L, ti)

2N×N

.

In this section, we consider the heat equation (2.1) with thegeneral conditions

(2.3) which can be separated into the 6 cases presented in (2.5)–(2.16). Applying the

boundary and the overdetermination conditions of these 6 cases results as follows.

2.3.1 Case 1

The Neumann heat flux boundary conditions (2.5) give

q¯=

−ux(0, tj)ux(L, tj)

2N

=

−µ1(tj)

µ2(tj)

2N

. (2.26)

Also, from (2.25) we obtain

h¯= B−1

(

Aq¯+ Cu

¯0+Dr

¯+ d

¯

)

. (2.27)

The overdetermination condition (2.6) can be rewritten as amatrix equation as follows:

[

V1 V2

]

h¯= k

¯, (2.28)

whereV1, V2 areN × N diagonal matrices of componentsv1(t1), . . . , v1(tN ) and

v2(t1), . . . , v2(tN), respectively, and k¯

is anN-column vector of the piecewise con-

stant approximation ofk(t), namely

k(t) = k(ti) =: ki, for t ∈ (tj−1, tj], j = 1, N.

Chapter 2. 25

Substituting (2.27) into (2.28) yields

[

V1 V2

]

B−1(

Aq¯+ Cu

¯0+Dr

¯+ d

¯

)

= k¯.

Rearranging this expression the inverse source problem in Case 1 reduces to solving

theN ×N linear system of equations

X1r¯= y

¯1, (2.29)

whereX1 =[

V1 V2

]

B−1D, and y¯1

= k¯−[

V1 V2

]

B−1(

Aq¯+ Cu

¯0+ d

¯

)

.

2.3.2 Case 2

We rearrange the matrix equation (2.25) as follows:

[

A0 AL

]


−[

B0 BL

]

u(0, tj)

u(L, tj)

+ Cu¯0

+Dr¯+ d

¯= 0

¯, (2.30)

whereA0 =

A0j(0, tj)

A0j(L, tj)

, AL =

ALj(0, tj)

ALj(L, tj)

, B0 =

B0j(0, tj) +12δij

B0j(L, tj)

,

andBL =

BLj(0, tj)

BLj(L, tj) +12δij

are2N × N matrices. Next, we apply the boundary

conditions (2.7) such that this system becomes

A0q¯0

+ ALµ¯2

−B0µ¯1

− BLh¯L

+ Cu¯0

+Dr¯+ d

¯= 0

¯,

whereµ¯1

=[

µ1(tj)]

N, µ

¯2=[

µ2(tj)]

N, q

¯0=[

q0j

]

N, and h

¯L=[

hLj

]

N. From this

system we obtain

−q¯0

h¯L

=[

A0 BL

]−1 (

ALµ¯2

− B0µ¯1

+ Cu¯0

+Dr¯+ d

¯

)

. (2.31)

Chapter 2. 26

The overdetermination condition (2.8) can be written in matrix equation as follows:

[

V1 V2

]

−q¯0

h¯L

= k¯. (2.32)

Substituting the expression (2.31) into (2.32) we obtain

[

V1 V2

] [

A0 BL

]−1 (

ALµ¯2

− B0µ¯1

+ Cu¯0

+Dr¯+ d

¯

)

= k¯.

Rearranging this expression the inverse source problem reduces to solving theN ×N

linear system of equations

X2r¯= y

¯2, (2.33)

where X2 =[

V1 V2

] [

A0 BL

]−1

D

and y¯2

= k¯−[

V1 V2

] [

A0 BL

]−1 (

ALµ¯2

− B0µ¯1

+ Cu¯0

+ d¯

)

.

2.3.3 Case 3

The Dirichlet boundary temperature conditions (2.9) give

h¯=

u(0, tj)

u(L, tj)

2N

=

µ1(tj)

µ2(tj)

2N

. (2.34)

Also, from (2.25) we obtain

q¯= A−1 (Bh

¯− Cu

¯0−Dr

¯− d

¯) . (2.35)

The overdetermination condition (2.10) can be written as

[

−V1 V2

]

q¯= k

¯. (2.36)

Chapter 2. 27

Substituting (2.35) into (2.36) gives theN ×N linear system of equations

X3r¯= y

¯3, (2.37)

whereX3 =[

−V1 V2

]

A−1D and y3 = −k¯+[

−V1 V2

]

A−1 (Bh¯− Cu

¯0− d

¯).

2.3.4 Case 4

Consider the boundary condition (2.11) which can be rewritten as

ux(L, t) = µ2(t)− v1(t)u(L, t).

Apply this to the matrix equation (2.30), derived from (2.25), then the system becomes

A0q¯+ AL(µ

¯2− V1h

¯L)− B0µ

¯1−BLh

¯L+ Cu

¯0+Dr

¯+ d

¯= 0

¯.

We rearrange the matrix equation above as follows:

[

A0 ALV1 +BL

]

−q¯0

h¯L

= ALµ¯2

− B0µ¯1

+ Cu¯0

+Dr¯+ d

¯.

From this system we obtain

−q¯0

h¯L

=[

A0 ALV1 +BL

]−1 (

ALµ¯2

− B0µ¯1

+ Cu¯0

+Dr¯+ d

¯

)

. (2.38)

The overdetermined condition (2.12) becomesux(0, t)+v2(t)µ2(t)−v2(t)v1(t)u(L, t) =k(t) and can be rewritten in the matrix form as,

[

I −V2V1]

−q¯0

h¯L

= k¯− V2µ

¯2, (2.39)

Chapter 2. 28

whereI is theN × N identity matrix. We then substitute (2.38) into (2.39) to obtain

theN ×N linear system of equations

X4r¯= y

¯4, (2.40)

where X4 =[

I −V2V1] [

A0 ALV1 +BL

]−1

D

and y¯4

= k¯−V2µ

¯2−[

I −V2V1] [

A0 ALV1 +BL

]−1 (

ALµ¯2

−B0µ¯1

+ Cu¯0

+ d¯

)

.

2.3.5 Case 5

Consider the boundary condition (2.13) which can be rewritten as

ux(L, t) = µ2(t)− v1(t)u(L, t). (2.41)

Apply this to the matrix equation (2.30), derived from (2.25), then the system becomes

−A0µ¯1

+ AL(µ¯2

− V1h¯L)− B0h

¯0−BLh

¯L+ Cu

¯0+Dr

¯+ d

¯= 0

¯.

We rearrange the matrix equation above as follows:

[

B0 ALV1 +BL

]

h¯= −A0µ

¯1+ ALµ

¯2+ Cu

¯0+Dr

¯+ d

¯.

From this system we obtain

h¯=[

B0 ALV1 +BL

]−1 (

−A0µ¯1

+ ALµ¯2

+ Cu¯0

+Dr¯+ d

¯

)

. (2.42)

The overdetermination condition (2.14) gives

[

I V2

]

h¯= k

¯. (2.43)

Chapter 2. 29

Substitute (2.42) into (2.43) to obtain theN ×N linear system of equations

X5r¯= y

¯5, (2.44)

where X5 =[

I V2

] [

B0 ALV1 +BL

]−1

D

and y¯5

= k¯−[

I V2

] [

B0 ALV1 +BL

]−1 (

−A0µ¯1

+ ALµ¯2

+ Cu¯0

+ d¯

)

.

2.3.6 Case 6

Consider the boundary condition (2.15) which can be rearranged as

q¯=


2N

=

−µ¯1

− V1u(0, tj)

µ¯2

− V2u(L, tj)

2N

.

Substituting this into the matrix equation (2.30) we obtain

A0(−µ¯1

− V1h¯0) + AL(µ

¯2− V2h

¯L)− B0h

¯0−BLh

¯L+ Cu

¯0+Dr

¯+ d

¯= 0

¯,

and this can be rearranged as

[

A0V1 +B0 ALV2 +BL

]

h¯= −A0µ

¯1+ ALµ

¯2+ Cu

¯0+Dr

¯+ d

¯.

Then we have

h¯=[

A0V1 +B0 ALV2 +BL

]−1 (

−A0µ¯1

+ ALµ¯2

+ Cu¯0

+Dr¯+ d

¯

)

. (2.45)

The overspecified condition (2.16) gives

[

V3 V4

]

h¯= k

¯. (2.46)

Chapter 2. 30

Substitute (2.45) into (2.46) to obtain theN ×N linear system of equations

X6r¯= y

¯6, (2.47)

where[

V3 V4

] [

A0V1 +B0 ALV2 +BL

]−1

D

and k¯−[

V3 V4

] [

A0V1 +B0 ALV2 +BL

]−1 (

−A0µ¯1

+ ALµ¯2

+ Cu¯0

+ d¯

)

.

From the above assembly one can see that the solution of the inverse heat source

problem (2.1)–(2.3) separated into the 6 cases, has been reduced to solving theN ×N

linear system of equations, generally written as

Xr¯= y

¯, (2.48)

whereX and y¯

are the coefficient matrix and the right-hand side vector, respectively,

corresponding to the case we are dealing with; that is, the linear systems of equations,

(2.29), (2.33), (2.37), (2.40), (2.44), and (2.47) for Cases 1–6, respectively. We note

that the system of equations (2.48) is ill-conditioned since the inverse problems un-

der investigation are ill-posed. Therefore, a straightforward inversion of (2.48) such

as the Gaussian elimination or the singular value decomposition will result into an

unstable numerical solution, especially when the right-hand side vector y¯

is contami-

nated by random noise as y¯ǫ = y

¯+ ǫ whereǫ represents the noise to contaminate into

the problem. In order to ensure a stable solution we employ the Tikhonov regulari-

sation method for (2.48) which gives solution (1.34) together with the regularisation

parameter chosen by the discrepancy principle. Let us denote byλdis the regularisa-

tion parameter which is determined by the discrepancy principle, i.e. the largestλ for

which the residual‖Xr¯− yǫ‖ becomes less than the noise levelǫ.

Chapter 2. 31

2.4 Numerical examples and discussion

In order to test the accuracy of the approximations, let us introduce the root mean

square error (RMSE) defined as

RMSE(r(t)) =

√

√

√

√

T

N

N∑

i=1

(

rexact(ti)− rnumerical(ti))2. (2.49)

2.4.1 Example 1

We consider a smooth benchmark test with the input data

u(x, 0) = u0(x) = 1 + x− x2,

f(x, t) = (1− x2)e−t, h(x, t) = (2 + x)et.

(2.50)

Assuming that all quantities involved have been non-dimensionalised we can takeT =

L = 1. In addition, the boundary and overdetermination conditions are as follows:

Case 1 ux(0, t) = et, ux(1, t) = −et, (2.51)

u(0, t) + u(1, t) = 2et. (2.52)

Case 2 u(0, t) = et, ux(1, t) = −et, (2.53)

ux(0, t) + u(1, t) = 2et. (2.54)

Case 3 u(0, t) = et, u(1, t) = et, (2.55)

etux(0, t) + tux(1, t) = (et − t)et, (2.56)

where v1(t) = et, v2(t) = t.

Case 4 u(0, t) = et, ux(1, t) + (1 + t)u(1, t) = tet, (2.57)

ux(0, t) + e−tux(1, t) = et − 1, (2.58)

where v1(t) = 1 + t, v2(t) = e−t.

Chapter 2. 32

Case 5 ux(0, t) = et, ux(1, t) + e−tu(1, t) = 1− et, (2.59)

u(0, t) + (1 + t)u(1, t) = (2 + t)et, (2.60)

where v1(t) = e−t, v2(t) = 1 + t.

Case 6 ux(0, t)− etu(0, t) = et − e2t, ux(1, t) + (1 + t)u(1, t) = tet, (2.61)

tu(0, t) + (1− t)u(1, t) = et, (2.62)

where v1(t) = et, v2(t) = 1 + t, v3(t) = t, v4(t) = 1− t.

In this example the analytical solution is given by

u(x, t) = (1 + x− x2)et, r(t) = e2t, (2.63)

for 0 ≤ x ≤ 1 and0 ≤ t ≤ 1. Note that the input data in expressions (2.50)–(2.62)

satisfy the conditions of Theorems 2.2.1–2.2.5 for the existence and uniqueness of the

solution of the inverse problems of Cases 1–6 under investigation.

Figure 2.1 and Table 2.1 show the condition numbers of matrixX in (2.48) cor-

responding toN0 = N ∈ 20, 40, 80 for all Cases 1–6. From these it can be seen

that the condition numbers of the matrixX for Cases 1, 5, and, 6 are high and this

ill-conditioning will need to be dealt with using the Tikhonov regularisation described

in Section 1.6.2. On the other hand, the matrices for Cases 2–4 do not have a very large

condition number and, in principle, the system of equations(2.48) can be solved di-

rectly using, for example, the Gauss elimination method or the SVD. In what follows,

we illustrate the numerical results obtained withN0 = N = 40.

Case 1

Figures 2.2(a)–2.2(c) show the analytical and numerical solutions forr(t), u(0, t) and

u(1, t), respectively, for exact input data and no regularisation,i.e. λ = 0. From

this figure it can be seen that although the solution for the boundary values of the

temperaturesu(0, t) andu(1, t) is stable and accurate, the retrieved source termr(t)

Chapter 2. 33

0 10 20 30 40 50 60 70 8010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Cond = 6.19E+3

Nor

mal

ised

Sin

gula

r V

alue

s

N

Cond = 1.26E+5

Cond = 7.49E+6

(a)

0 10 20 30 40 50 60 70 8010

−3

10−2

10−1

100

Cond = 87

Nor

mal

ised

Sin

gula

r V

alue

s

N

Cond = 248

Cond = 705

(b)

0 10 20 30 40 50 60 70 8010

−3

10−2

10−1

100

Cond = 22

Nor

mal

ised

Sin

gula

r V

alue

s

N

Cond = 54

Cond = 125

(c)

0 10 20 30 40 50 60 70 8010

−3

10−2

10−1

100

Cond = 38

Nor

mal

ised

Sin

gula

r V

alue

s

N

Cond = 107

Cond = 298

(d)

0 10 20 30 40 50 60 70 8010

−10

10−8

10−6

10−4

10−2

100

Cond = 1.52E+4

Nor

mal

ised

Sin

gula

r V

alue

s

N

Cond = 7.36E+5

Cond = 1.76E+8

(e)

0 10 20 30 40 50 60 70 8010

−10

10−8

10−6

10−4

10−2

100

Cond = 1.61E+4

Nor

mal

ised

Sin

gula

r V

alue

s

N

Cond = 2.52E+6

Cond = 5.89E+9

(f)

Figure 2.1: The normalised singular values of matrixX for (a) Case 1 – (f) Case 6 forN0 = N = 20 (− · −), 40 (· · · ), 80 (−−−), for Example 1.

Chapter 2. 34

Table 2.1: The condition numbers of the matrixX in equation (2.48) forN = N0 ∈20, 40, 80 for Example 1 Cases 1–6.

CaseCondition Number

N = N0 = 20 N = N0 = 40 N = N0 = 801 6.19E+3 1.26E+5 7.49E+62 87 248 7053 22 54 1254 38 107 2985 1.52E+4 7.36E+5 1.76E+86 1.61E+4 2.52E+6 5.89E+9

seems unstable. This is to be expected since the inverse problem in Case 1 is ill-

posed, see also the condition number in Table 2.1. Regularisation needs to be employed

and more stable results are illustrated in Figure 2.3. In this figure, numerical results

obtained with the zeroth-order Tikhonov regularisation. Initially, we have tried the

L-curve criterion, but we have found that anL-corner could not be clearly identified.

We then have tried with the trial and error and found that the regularisation parameters

λ ∈ 10−7, 10−5 are most suitable as presented in Figure 2.3. Moreover, although

not illustrated, it is reported that the slight inaccuracies neart = 1 observed in Figure

2.3(a) can be further eliminated by employing higher-orderregularisations such as the

first- or second-order.

Table 2.2: The RMSE for the zeroth- and first-order Tikhonov regularisation forp ∈0, 1, 3% noise, for Example 1 Case 1.

Regularisation p(%)parameter RMSE

λ r(t) u(0, t) u(1, t)- 0 0 0.983 3.67E-4 3.67E-4- 1 0 4.90E+2 1.90E-1 1.81E-1

zeroth-order0 λ=1.0E-5 0.499 1.27E-3 7.92E-51 λdis=6.6E-4 1.264 2.01E-2 7.31E-33 λdis=4.0E-3 1.982 6.25E-2 2.91E-2

first-order0 λ=1.0E-7 9.49E-3 3.06E-5 3.07E-51 λdis=4.3E-2 0.364 7.19E-3 3.03E-33 λdis=9.7E-1 1.007 3.59E-2 1.78E-2

In order to investigate the stability of the numerical solution we add noise to the

Chapter 2. 35

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

9

10

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(c)

Figure 2.2: The analytical (—–) and numerical (− · −) results of (a)r(t), (b) u(0, t),and (c)u(1, t) for exact data andλ = 0, for Example 1 Case 1.

right-hand side of the overspecified condition (2.6) as

k¯ǫ = k

¯+ random(′Normal′, 0, σ, 1, N), (2.64)

where therandom(′Normal′, 0, σ, 1, N) is a command in MATLAB which generates

the random variables by normal distribution with zero mean and standard deviationσ,

computed as

σ = p× maxt∈[0,1]

|k(t)|, (2.65)

wherep represents the percentage of noise. In Case 1,k is given by (2.52) and there-

fore,σ = 2ep in (2.65). The ill-posedness of the inverse problem and the instability of

Chapter 2. 36

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(b)

Figure 2.3: The analytical (—–) and numerical (− · −) results ofr(t) obtained byusing the zeroth-order Tikhonov regularisation with the regularisation parameters (a)λ = 10−7 gives RMSE=0.319, and (b)λ = 10−5 gives RMSE=0.499, for exact datafor Example 1 Case 1.

the numerical solution in Case 1 are further enhanced by the presence of noise in the

measured data, as illustrated in Figure 2.4 (compare with Figure 2.2). In order to re-

move the highly unwanted oscillations recorded in Figure 2.4, we employ the Tikhonov

regularisation with the choice of the regularisation parameterλ, as described in (1.34),

given by the discrepancy principle as introduced in Section1.6.3. Figures 2.5(a) and

2.6(a) present the discrepancy principle curves obtained by the Tikhonov regularisa-

tion of order zero and one, respectively, forp = 1% and3% noisy data. Generated

as in (2.64), this results in the amounts of noiseǫ = 0.32 and 1.01 forp = 1% and

3%, respectively. The intersections between these horizontal lines and the discrepancy

(residual) curves yield the regularisation parameter denoted byλdis and tabulated in

Table 2.2. With these values ofλdis, Figures 2.5(b)–2.5(d) and 2.6(b)–2.6(d) present

the numerical results forr(t), u(0, t), andu(1, t) obtained using the zeroth- and first-

order Tikhonov regularisation, respectively. By comparing Figures 2.5(b) and 2.6(b)

it can be seen that the first-order regularisation produces more accurate and stable re-

sults than the zeroth-order regularisation since it imposes a higher-order smoothness

constraint onto the solution.

Chapter 2. 37

0 0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500

1000

1500

2000

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

t

u(1

,t)

(c)

Figure 2.4: The analytical (—–) and numerical (− · −) results of (a)r(t), (b) u(0, t),and (c)u(1, t) for p = 1% noisy data andλ = 0, for Example 1 Case 1.

Cases 2–4

In Cases 2–4, Table 2.1 shows that the condition numbers of the matricesX2, X3, and

X4 are much smaller than the condition numbers of the matrices for the rest of the

cases. Therefore, for exact data, i.e.p = 0, we expect accurate and stable results

of the inversion even if no regularisation is employed, i.e.λ = 0. This is clearly

illustrated in Figure 2.7 for Case 2 where it can be seen that the agreement between

the analytical and numerical solutions forr(t), u(1, t) andux(0, t) is excellent. The

same overlapping agreement has also been obtained for Cases3 and 4 and therefore

these results are omitted. Next we addp% noise in the input data, as described in

Chapter 2. 38

10−6

10−5

10−4

10−3

10−2

10−1

100

10−2

10−1

100

101

ε = 0.32λ = 6.6E−4

ε = 1.01λ = 4.0E−3

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(d)

Figure 2.5: (a) The discrepancy principle curve, and the analytical (—–) and numericalresults of (b)r(t), (c)u(0, t), and (d)u(1, t) obtained using the zeroth-order Tikhonovregularisation forp = 1% (− · −) andp = 3% (− − −) noise with the regularisationparametersλdis given in Table 2.2, for Example 1 Case 1.

(2.64). According to expression (2.65), and equations (2.54), (2.56), and (2.58), we

have the standard deviationsσ = 2ep, σ = (e2 − e)p, andσ = (e − 1)p for Cases

2–4, respectively. As expected, and previously reported inFigure 2.4 for Case 1, if

no regularisation is imposed, i.e.λ = 0, whenp = 1% noise contaminates the input

datak(t), then an unstable solution forr(t) is obtained, see Figures 2.8, 2.11, and

2.14 for Cases 2–4, respectively. However, the high oscillations in these figures have

smaller magnitudes than these reported in Figure 2.4 for Case 1. This is consistent

with the fact that the inverse problems of Cases 2–4 are less ill-posed than the inverse

Chapter 2. 39

10−4

10−3

10−2

10−1

100

101

102

103

0.5

1

1.5

2

ε = 0.32λ = 4.3E−2

ε = 1.01λ = 9.7E−1

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(d)

Figure 2.6: (a) The discrepancy principle curve, and the analytical (—–) and numericalresults of (b)r(t), (c) u(0, t), and (d)u(1, t) obtained using the first-order Tikhonovregularisation forp = 1% (− · −) andp = 3% (− − −) noise with the regularisationparametersλdis given in Table 2.2, for Example 1 Case 1.

problem of Case 1 (and Cases 5 and 6), see the condition numbers reported in Table

2.1. It is also interesting to remark that the retrieval of the boundary temperature

is more accurate and stable than of the heat flux, e.g. compareFigures 2.8(b) and

2.14(b) with Figures 2.8(c) and 2.14(c), see also Figures 2.11(b) and 2.11(c). This is

to be expected since retrieving higher-order derivatives is less accurate than retrieving

lower-order derivatives, see Lesnic et al. [38]. In order toobtain a stable solution,

regularisation needs to be employed. The numerical resultsobtained forp ∈ 1, 3%noise using the zeroth-order Tikhonov regularisation withthe regularisation parameter

Chapter 2. 40

chosen according to the discrepancy principle are shown in Figures 2.9, 2.12, and

2.15 for Cases 2–4, respectively. The corresponding results obtained using the first-

order regularisation illustrated in Figures 2.10, 2.13, and 2.16 show much improvement

in terms of both stability and accuracy compared to the results obtained using the

zeroth-order regularisation. The regularisation parameters and the RMSE errors of the

output solutions are given in Tables 2.3–2.5 for Cases 2–4, respectively. From these

tables it can be observed that, as expected,λdis and RMSE increase with increasing the

percentage of noisep.

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(c)

Figure 2.7: The analytical (—–) and numerical (− · −) results of (a)r(t), (b) u(1, t),and (c)ux(0, t) for exact data andλ = 0, for Example 1 Case 2.

Chapter 2. 41

0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

6

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(c)

Figure 2.8: The analytical (—–) and numerical (− · −) results of (a)r(t), (b) u(1, t),and (c)ux(0, t) for p = 1% noisy data andλ = 0, for Example 1 Case 2.

Cases 5 and 6

In Cases 5 and 6 we expect even higher ill-conditioning to occur in the systems of equa-

tions (2.44) and (2.47), compare the condition numbers in Table 2.1. This is reflected

indeed in the numerical results presented in Figures 2.17(a) and 2.18 for Case 5, and

even more prominently in Figure 2.21 for Case 6 where unstable numerical results can

be clearly seen ifλ = 0 even when exact data are inverted. Whenp% noise is added

to the measured data (2.60) and (2.62), the standard deviations in (2.65) are given by

σ = 3ep andσ = ep for Cases 5 and 6, respectively. Numerical results obtainedusing

the zeroth- and first-order regularisation are presented inFigures 2.19, 2.20 and Table

Chapter 2. 42

10−6

10−5

10−4

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

101

ε = 0.34λ = 2.8E−3

ε = 1.02λ = 8.8E−3

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(c)

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(d)

Figure 2.9: (a) The discrepancy principle curve, and the analytical (—–) and numericalresults of (b)r(t), (c)u(1, t), and (d)ux(0, t) obtained using the zeroth-order Tikhonovregularisation forp = 1% (− · −) andp = 3% (− − −) noise with the regularisationparametersλdis given in Table 2.3, for Example 1 Case 2.


Regularisation p(%)parameter RMSEλdis r(t) u(1, t) ux(0, t)

- 0 0 7.15E-3 4.33E-5 4.33E-5- 1 0 3.29 5.94E-3 5.31E-2

zeroth-order1 2.8E-3 0.785 7.63E-3 5.83E-23 8.8E-3 1.331 1.66E-2 1.24E-1

first-order1 3.0E-1 3.23E-1 3.29E-3 2.95E-23 9.4E-1 5.53E-1 8.58E-3 5.81E-2

Chapter 2. 43

10−4

10−3

10−2

10−1

100

101

102

103

104

105

100

ε = 0.34λ = 3.0E−1

ε = 1.02λ = 9.4E−1

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(d)

Figure 2.10: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), (c) u(1, t), and (d)ux(0, t) obtained using the first-orderTikhonov regularisation forp = 1% (− · −) andp = 3% (− − −) noise with theregularisation parametersλdis given in Table 2.3, for Example 1 Case 2.

Chapter 2. 44

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

6

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(b)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(c)

Figure 2.11: The analytical (—–) and numerical (−·−) results of (a)r(t), (b)ux(0, t),and (c)ux(1, t) for p = 1% noisy data andλ = 0, for Example 1 Case 3.

2.6 for Case 5, whilst for Case 6 the corresponding results are presented in Figures

2.22, 2.23 and Table 2.7. Furthermore, for Case 6, which is the most ill-conditioned

case, we have increased the percentage of noise top = 5% in order to show that the

Tikhonov regularisation method combined with the BEM can satisfactorily deal in a

stable and accurate manner with higher measurement errors.We finally report that,

although not illustrated, we have also implemented the second-order Tikhonov reg-

ularisation and similar results, in terms of accuracy and stability, to those given by

first-order regularisation have been obtained.

Chapter 2. 45

0 0.0001 0.001 0.01 0.1 1 10 100 1000 1000010

−3

10−2

10−1

100

101

ε = 0.29λ = 3.2E−3

ε = 0.84λ = 8.7E−3

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(d)

Figure 2.12: (a) The discrepancy principle curve, and the analytical (—–) and numer-ical results of (b)r(t), (c) ux(0, t), and (d)ux(1, t) obtained using the zeroth-orderTikhonov regularisation forp = 1% (− · −) andp = 3% (− − −) noise with theregularisation parametersλdis given in Table 2.4, for Example 1 Case 3.

2.4.2 Example 2

We finally investigate specially in the most ill-posed Case 6, the retrieval of a non-

smooth heat source given by

r(t) =

∣

∣

∣

∣

t− 1

2

∣

∣

∣

∣

+ 1, 0 < t < 1 = T, (2.66)

Chapter 2. 46

10−4

10−3

10−2

10−1

100

101

102

103

104

105

10−2

10−1

100

101

ε = 0.29λ = 3.0E−1

ε = 0.84λ = 5.8E−1

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

t

ux(1

,t)

(d)

Figure 2.13: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), (c) ux(0, t), and (d)ux(1, t) obtained using the first-orderTikhonov regularisation forp = 1% (− · −) andp = 3% (− − −) noise with theregularisation parametersλdis given in Table 2.4, for Example 1 Case 3.

and, for brevity, the boundary and overdetermination conditions as given by (2.61),

(2.62) and the input data as follows:

u(x, 0) = u0(x) = 1 + x− x2, f(x, t) = (1− x2)e−t,

h(x, t) = (3 + x− x2)et − (1− x2)

(∣

∣

∣

∣

t− 1

2

∣

∣

∣

∣

+ 1

)

e−t,(2.67)

which are generated from the analytical temperature solution

u(x, t) = (1 + x− x2)et. (2.68)

Chapter 2. 47


Regularisation p(%)parameter RMSEλdis r(t) ux(0, t) ux(1, t)

- 0 0 3.29E-3 4.04E-5 1.49E-4- 1 0 1.776 3.58E-2 2.41E-2

zeroth-order1 3.2E-3 4.42E-1 3.65E-2 1.74E-23 8.7E-3 7.80E-1 6.81E-2 3.09E-2

first-order1 3.0E-1 1.78E-1 1.72E-2 7.58E-33 5.8E-1 2.47E-1 2.41E-2 1.07E-2

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

9

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(d)

Figure 2.14: The analytical (—–) and numerical (− · −) results of (a)r(t), (b)u(1, t),(c) ux(0, t), andux(1, t) for p = 1% noisy data andλ = 0, for Example 1 Case 4.

Numerical results are presented forN = N0 = 80. Also, results are illustrated for the

first-order Tikhonov regularisation only, as for the zeroth-order regularisation the re-

Chapter 2. 48


Regularisation p(%)parameter RMSEλdis r(t) u(1, t) ux(0, t) ux(1, t)

- 0 0 4.13E-3 3.63E-5 3.19E-5 5.77E-5- 1 0 4.98E-1 9.11E-4 1.71E-2 1.33E-3

zeroth-order1 6.2E-4 4.35E-1 1.52E-3 1.89E-2 2.70E-33 2.0E-3 7.31E-1 4.55E-3 4.95E-2 8.24E-3

first-order1 4.8E-2 1.23E-1 9.29E-4 9.61E-3 1.38E-33 1.3E-1 1.77E-1 2.16E-3 1.72E-2 3.15E-3


Regularisation p(%)parameter RMSEλdis r(t) u(0, t) u(1, t) ux(1, t)

- 0 0 5.598 3.35E-3 1.72E-3 6.76E-4- 1 0 2.9E+3 1.776 9.09E-1 3.56E-1

zeroth-order1 2.2E-3 1.584 3.92E-2 1.60E-2 7.79E-33 9.5E-3 2.033 9.38E-2 4.73E-2 2.53E-2

first-order1 1.7E-1 0.673 2.29E-2 1.02E-2 4.93E-33 3.4 1.334 7.52E-2 3.92E-2 2.00E-2

Table 2.7: The RMSE for the zeroth- and first-order Tikhonov regularisation forp ∈0, 1, 3, 5% noise, for Example 1 Case 6.

Regularisation p(%)parameter RMSEλdis r(t) u(0, t) u(1, t) ux(0, t) ux(1, t)

- 0 0 1.49E+2 2.58E-2 6.08E-2 4.66E-2 1.09E-1- 1 0 6.78E+4 1.17E+1 2.77E+1 2.13E+1 4.98E+1

zeroth-order1 1.8E-4 1.380 1.96E-2 9.67E-3 4.59E-2 1.68E-23 6.3E-4 1.559 4.03E-2 2.61E-2 8.14E-2 4.28E-25 8.8E-4 1.737 4.07E-2 2.23E-2 8.51E-2 3.78E-2

first-order1 2.4E-2 5.90E-1 1.19E-2 6.82E-3 2.59E-2 1.10E-23 8.4E-2 5.36E-1 1.71E-2 1.12E-2 2.81E-2 1.52E-25 9.5E-2 5.04E-1 1.80E-2 1.36E-2 3.49E-2 2.31E-2

Chapter 2. 49

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

ε = 0.10λ = 6.1E−4

ε = 0.31λ = 2.0E−3

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(c)

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(d)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

t

ux(1

,t)

(e)

Figure 2.15: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), (c) u(1, t), (d) ux(0, t), and (e)ux(1, t) obtained using thezeroth-order Tikhonov regularisation forp = 1% (− · −) andp = 3% (− − −) noisewith the regularisation parametersλdis given in Table 2.5, for Example 1 Case 4.

Chapter 2. 50

0.0001 0.001 0.01 0.1 1 10 100 1000 1000010

−3

10−2

10−1

100

101

ε = 0.10λ = 4.1E−2

ε = 0.31λ = 1.3E−1

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(d)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

t

ux(1

,t)

(e)

Figure 2.16: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), (c) u(1, t), (d) ux(0, t), and (e)ux(1, t) obtained using thefirst-order Tikhonov regularisation forp = 1% (− · −) andp = 3% (− − −) noisewith the regularisation parametersλdis given in Table 2.5, for Example 1 Case 4.

Chapter 2. 51

0 0.2 0.4 0.6 0.8 1−15

−10

−5

0

5

10

15

20

25

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(c)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(d)

Figure 2.17: The analytical (—–) and numerical (− · −) results of (a)r(t), (b)u(0, t),(c) u(1, t) and (d)ux(1, t) for exact data andλ = 0, for Example 1 Case 5.

Chapter 2. 52

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1x 10

4

t

r(t

)

x 104

(a)

0 0.2 0.4 0.6 0.8 1−5

0

5

10

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

7

t

u(1

,t)

(c)

0 0.2 0.4 0.6 0.8 1−4

−3.5

−3

−2.5

−2

−1.5

−1

t

ux(1

,t)

(d)

Figure 2.18: The analytical (—–) and numerical (− · −) results of (a)r(t), (b) u(0, t),(c) u(1, t) and (d)ux(1, t) for p = 1% noisy data andλ = 0, for Example 1 Case 5.

Chapter 2. 53

0 0.0001 0.001 0.01 0.1 1 10 100 1000 10000

100

101

ε = 0.50λ = 2.2E−3

ε = 1.52λ = 9.5E−3

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(d)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

t

ux(1

,t)

(e)

Figure 2.19: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), (c) u(0, t), (d) u(1, t), and (e)ux(1, t) obtained using thezeroth-order Tikhonov regularisation forp = 1% (− · −) andp = 3% (− − −) noisewith the regularisation parametersλdis given in Table 2.6, for Example 1 Case 5.

Chapter 2. 54

10−3

10−2

10−1

100

101

102

103

104

105

106

11

2

ε = 0.50λ = 2.7E−1

ε = 1.52

λ = 3.4

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(d)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(e)

Figure 2.20: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), (c) u(0, t), (d) u(1, t), and (e)ux(1, t) obtained using thefirst-order Tikhonov regularisation forp = 1% (− · −) andp = 3% (− − −) noisewith the regularisation parametersλdis given in Table 2.6, for Example 1 Case 5.

Chapter 2. 55

0 0.2 0.4 0.6 0.8 1−300

−200

−100

0

100

200

300

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(d)

0 0.2 0.4 0.6 0.8 1−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(e)

Figure 2.21: The analytical (—–) and numerical (− · −) results of (a)r(t), (b)u(0, t),(c) u(1, t), (d) ux(0, t), and (e)ux(1, t) for exact data andλ = 0, for Example 1 Case6.

Chapter 2. 56

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

10−2

10−1

100

101

ε = 0.17λ = 1.8E−4

ε = 0.47λ = 6.3E−4

ε = 0.85λ = 8.8E−4

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(d)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(e)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(f)

Figure 2.22: (a) The discrepancy principle curve, and the analytical (—–) and numeri-cal results of (b)r(t), (c)u(0, t), (d)u(1, t), (e)ux(0, t), and (f)ux(1, t) obtained usingthe zeroth-order Tikhonov regularisation forp ∈ 1(−·−), 3(· · · ), 5(−−−)% noisewith the regularisation parametersλdis given in Table 2.7, for Example 1 Case 6.

Chapter 2. 57

10−4

10−3

10−2

10−1

100

101

102

103

104

105

10−0.8

10−0.6

10−0.4

10−0.2

100

ε = 0.17λ = 2.4E−2

ε = 0.47λ = 8.4E−2

ε = 0.85λ = 9.5E−2

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(1

,t)

(d)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

ux(0

,t)

(e)

0 0.2 0.4 0.6 0.8 1−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

t

ux(1

,t)

(f)

Figure 2.23: (a) The discrepancy principle curve, and the analytical (—–) and numeri-cal results of (b)r(t), (c)u(0, t), (d)u(1, t), (e)ux(0, t), and (f)ux(1, t) obtained usingthe first-order Tikhonov regularisation forp ∈ 1(− · −), 3(· · · ), 5(−−−)% noisewith the regularisation parametersλdis given in Table 2.7, for Example 1 Case 6.

Chapter 2. 58

sults obtained were oscillatory as in Figure 2.22(b). Figure 2.24 shows the numerical

results obtained for various amounts of noisep ∈ 0, 0.1, 0.5, 1% for the regular-

isation parametersλdis given in Table 2.8. From this figure it can be seen that the

regularised BEM can invert accurately up to about0.1% noisy data. For higher levels

of noise the reconstruction of the non-smooth heat source (2.66) starts to deteriorate.

Table 2.8: The RMSE for the first-order Tikhonov regularisation for p ∈0, 0.1, 0.5, 1% noise, for Example 2.

p(%)parameter RMSE

λ r(t) u(0, t) u(1, t) ux(0, t) ux(1, t)0 λ=1.0E-5 1.63E-3 1.40E-5 9.88E-6 1.77E-5 1.54E-5

0.1 λdis=2.7E-2 2.60E-2 1.11E-3 7.93E-4 1.48E-3 1.02E-30.5 λdis=3.1E-1 1.02E-1 4.51E-3 3.11E-3 6.84E-3 4.21E-31 λdis=1.1 1.28E-1 8.10E-3 5.84E-3 1.09E-2 7.61E-3

10−4

10−3

10−2

10−1

100

101

102

103

10−2

10−1

100

ε = 0.023λ = 2.7E−2

ε = 0.11λ = 3.1E−1

ε = 0.24λ = 1.1

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

t

r(t

)

(b)

Figure 2.24: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t) obtained using the first-order Tikhonov regularisation forp ∈ 0( ), 0.1(− · −), 0.5(· · · ), 1(− − −)% noise with the regularisation pa-rametersλdis given in Table 2.8, for Example 2.

2.5 Conclusions

In this chapter, inverse problems with nonlocal boundary conditions have been inves-

tigated in order to find the time-dependent heat source and the temperature entering

Chapter 2. 59

equation (2.1). Three general boundary and overdetermination conditions (2.3) have

been expanded into 6 separate Cases 1–6 generating six inverse problems to solve in

the context of inverse time-dependent source identification. It was found, see Table

2.1, that most ill-posed problems are given by Cases 6 and 5 followed by Case 1. It

turns out that Cases 2–4 are less ill-posed compared to the previous ones. This is con-

sistently reflected in the accuracy and stability of the numerical results obtained with

no regularisation for both exact and noisy data. Regularisation was found essential

in all cases investigated, with the first-order regularisation performing better than the

zeroth-order one.

As we have studied in this chapter, the nonlocal boundary andoverdetermination

conditions are considered in general boundary condition form. In the next chapter, the

nonlocal boundary condition will be considered together with an integral overdetermi-

nation to find the time-dependent heat source.

Chapter 3. 60

Chapter 3


Heat Source from Nonlocal Boundary

and Integral Conditions

3.1 Introduction

In many of studies of solving inverse source problems for theheat equation, see e.g.

[15, 17, 29, 61, 63, 66–68], the overdetermination conditions were selected among

classical boundary conditions and similar conditions given at interior points located

inside the body under investigation. More general, nonlocal and integral overdetermi-

nation conditions have also been considered, see the monographs [25] and [43].

In the previous chapter, we have investigated the retrievalof the time-dependent

heat sourcer(t) and the temperatureu(x, t) in the heat equation (2.1) with the three

general boundary and overdetermination conditions (2.3).In this chapter, the inves-

tigation of finding the time-dependent heat source and the temperature for the heat

equation is still of our interest, but an integral conditionis considered as overdetermi-

nation condition.

This chapter is organised as follows. In Section 3.2, the mathematical inverse for-

61

Chapter 3. 62

mulation is described and the numerical discretisation of the problem using the BEM

is presented in Section 3.3. The TSVD and the Tikhonov regularisation are described

in Section 3.4, as procedures for overcoming the instability of the solution. Finally,

Sections 3.5 and 3.6 discuss the numerical results and highlight the conclusions of this

chapter.


Consider the problem of finding the time-dependent heat sourcer(t) ∈ C([0, T ]) and

the temperatureu(x, t) ∈ C2,1(DT ) ∩ C1,0(DT ) which satisfy the heat conduction

equation

ut = uxx + r(t)f(x, t), (x, t) ∈ DT , (3.1)

whereL = 1 in the definition ofDT in (1.1), subject to the initial condition (1.7),

namely

u(x, 0) = u0(x), x ∈ [0, 1], (3.2)

the boundary conditions

u(0, t) = u(1, t), ux(0, t) + αu(0, t) = 0, t ∈ [0, T ], (3.3)

and the energy/mass specification

∫ 1

0

u(x, t)dx = E(t), t ∈ [0, T ], (3.4)

whereα 6= 0 is a given constant andf , u0, E are given functions. The first periodic-

ity condition in (3.3) is nonlocal, whilst (3.4) specifies anintegral specification of the

energy of the thermal system. The nature of the boundary conditions (3.3) in mathe-

matical biology is demonstrated in [41], whilst the prescription of the energy, or mass,

(3.4), is encountered in heat transfer applications, [10, 22].

The unique solvability of the inverse problem (3.1)–(3.4) has been established in

Chapter 3. 63

[18], as given by the following theorem.

Theorem 3.2.1 Let the following conditions be satisfied:

(A1) u0(x) ∈ C2[0, 1]; u0(0) = u0(1), u′0(0) + αu0(0) = 0;

(A2) E(t) ∈ C1[0, T ]; E(0) =∫ 1

0u0(x)dx;

(A3) f(x, t) ∈ C(DT ); f(·, t) ∈ C2[0, 1], ∀t ∈ [0, T ]; f(0, t) = f(1, t),

fx(0, t) + αf(0, t) = 0; and∫ 1

0f(x, t)dx 6= 0, ∀t ∈ [0, T ].

Then the inverse problem(3.1)–(3.4)has a unique solution(r(t), u(x, t)) ∈ C[0, T ]×(C2,1(DT ) ∩ C1,0(DT )).

Although the inverse problem (3.1)–(3.4) is uniquely solvable, it is still ill-posed. In

the next section we will demonstrate how the BEM discretising numerically, the heat

equation (3.1) can be used together with the regularisation, to be described in Section

3.4 either the TSVD or the Tikhonov regularisation, in orderto obtain stable solutions.


In this section, we use the BEM to discretise the heat equation (3.1). As introduced in

Section 1.3, the use of BEM results in the boundary integral equation

η(x)u(x, t) =

∫ t

0

[

G(x, t, ξ, τ)∂u

∂n(ξ)(ξ, τ)− u(ξ, τ)

∂G

∂n(ξ)(x, t, ξ, τ)

]

ξ∈0,1

dτ

+

∫ 1

0

G(x, t, y, 0)u(y, 0) dy+

∫ 1

0

∫ t

0

G(x, t, y, τ)r(τ)f(y, τ) dτdy,

(x, t) ∈ [0, 1]× (0, T ]. (3.5)

Chapter 3. 64

Using the same discretisation as described in Section 2.3, we obtain

η(x)u(x, t) =

N∑

j=1


+

N0∑

k=1

Ck(x, t)u0,k + d(x, t), (3.6)

where source functionsr(t) andf(x, t) are assumed to be piecewise constant approx-

imations as defined in (2.21). Then, the double integral termis approximated as

d(x, t) =

∫ t

0

r(τ)

(∫ 1

0

G(x, t, y, τ)f(y, τ) dy

)

dτ =

N∑

j=1

Dj(x, t)rj ,

where

Dj(x, t) =

∫ tj

tj−1

∫ 1

0

G(x, t, y, τ)f(y, tj)dydτ =

∫ 1

0


is calculated using the midpoint integration rule. Here, the integral equation (3.6) can

be written as

η(x)u(x, t) =N∑

j=1


+

N0∑

k=1

Ck(x, t)u0,k +N∑

j=1

Dj(x, t)rj. (3.7)

By applying (3.7) at the boundary nodes(0, ti) and(1, ti) for i = 1, N , we obtain the

system of2N equations

Aq¯−Bh

¯+ Cu

¯0+Dr

¯= 0

¯, (3.8)

Chapter 3. 65

where all matrices and vectors are defined the same as in (2.25). From the boundary

conditions (3.3), we can express the boundary temperature h¯

as

h¯=

h0j

hLj

=

u(0, tj)

u(1, tj)

=

− 1αux(0, tj)

u(0, tj)

=

− 1αux(0, tj)

− 1αux(0, tj)

=1

α

q0j

q0j

.

Then, the system of equations (3.8) can be rewritten as

(

A− 1

α(B +B∗)

)

q¯+ Cu

¯0+Dr

¯= 0

¯, (3.9)

whereB∗ =

BLj(0, ti) −BLj(0, ti)

BLj(1, ti) +12δij −BLj(1, ti)− 1

2δij

2N×2N

. Then, we can express

the flux q¯

as

q¯= −

(

A− 1

α(B +B∗)

)−1

(Cu¯0

+Dr¯) . (3.10)

We now discretise the integral expression in (3.4), via the midpoint numerical in-

tegral approximation, as

∫ 1

0

u(x, t) dx =1

N0

N0∑

k=1

u(xk, t).

Applying this att = ti for i = 1, N , we obtain

1

N0

N0∑

k=1

u(xk, ti) =

∫ 1

0

u(x, ti) dx = E(ti) =: ei for i = 1, N. (3.11)

Using the integral equation (3.5), as before, equation (3.11) results in the system ofN

equations

1

N0

N0∑

k=1

[(

A(1)k − 1

α(B

(1)k +B

(1)k

∗)

)

q¯+ C

(1)k u

¯0+D

(1)k r

¯

]

= E¯, (3.12)

Chapter 3. 66

where

A(1)k =

[

A0j(xk, ti) ALj(xk, ti)]

N×2N, B

(1)k =

[

B0j(xk, ti) BLj(xk, ti)]

N×2N,

B(1)k

∗=[

BLj(xk, ti) −BLj(xk, ti)]

N×2N, C

(1)k =

[

Cl(xk, ti)]

N×N0

,

and D(1)k =

[

Dj(xk, ti)]

N×N, E

¯=[

ei

]

N, for k, l = 1, N0, i, j = 1, N.

Substituting the expression (3.10) into (3.12), we obtain asystem ofN ×N linear

equations as follows:

Xr¯= y

¯, (3.13)

where X =1

N0

N0∑

k=1

[

−(

A(1)k − 1

α(B

(1)k +B

(1)k

∗)

)(

A− 1

α(B +B∗)

)−1

D +D(1)k

]

and y¯= E

¯+

1

N0

N0∑

k=1

[

(

A(1)k − 1

α(B

(1)k +B

(1)k

∗)

)(

A− 1

α(B +B∗)

)−1

C − C(1)k

]

u¯0

.

Since the problem is ill-posed, then the system of equations(3.13) is ill-conditioned.

In the next section, we will deal with this ill-conditioningusing regularisation in order

to obtain a stable solution.

3.4 Regularisation

As we have mentioned before, the inverse time-dependent heat source problem (3.1)–

(3.4) has a unique solution. However this inverse problem isstill ill-posed since small

errors into the energy measurement (3.4) result in large errors in the solution r¯. In

order to model this, we investigate the stability of the numerical solution with respect

to noise in the energy data (3.4), defined as

E¯ǫ = E

¯+ random(′Normal′, 0, σ, 1, N), (3.14)

Chapter 3. 67

with the standard deviation computed by

σ = p× maxt∈[0,T ]

|E(t)|. (3.15)

When the noise is present the right-hand side of equation (3.13) is contaminated by

some noiseǫ,

‖y¯ǫ − y

¯‖ ≈ ǫ, (3.16)

therefore we have to solve the following system of linear equations instead of (3.13):

Xr¯= y

¯ǫ, (3.17)

then the inverse solution r¯= X−1y

¯ǫ of (3.13) will be a poor unstable approximation to

the exact solution. This instability can be overcome by employing the regularisation

method, and this study, we utilise either the TSVD or the Tikhonov regularisation

methods to solve the linear and ill-conditioned system of equations (3.17).

When the noise is present, we employ the TSVD procedure, as described in Sec-

tion 1.6.1 by using the[U,Σ,V]=svds(X,Nt) command in MATLAB where theNt is the

truncated level indicated by theL-curve method or the discrepancy principle. Alterna-

tively, the regularisation namely the Tikhonov regularisation is another way to obtain

a stable solution, as described in Subsection 1.6.2, yielding the regularised solution

(1.34).


In this section, we present a couple of benchmark test examples to illustrate the accu-

racy and stability of the BEM combined with the Tikhonov regularisation technique

presented in Section 3.4. In order to illustrate the accuracy of the numerical results,

the RMSE defined in (2.49) is also used here.

Chapter 3. 68

3.5.1 Example 1

In the first example, we consider the inverse problem (3.1)–(3.4) withT = 1, the input

data

u(x, 0) = u0(x) = 1 + x− x2, f(x, t) = (3 + x− x2)e−t,∫ 1

0

u(x, t) dx = E(t) =7et

6,

(3.18)

and subject to the boundary condition (3.3) withα = −1. Then the analytical solution

of the inverse problem (3.1)–(3.4) is given by

u(x, t) = (1 + x− x2)et, r(t) = e2t. (3.19)

The normalised singular values of matrixX in equation (3.13) forN0 = N ∈20, 40, 80 are shown in Figure 3.1. The corresponding condition numbers of matrix

X are 248, 780, and 2393 forN0 = N ∈ 20, 40, 80, respectively. These values

indicate that the system of equations (3.13) is mildly ill-conditioned [42].

0 10 20 30 40 50 60 70 8010

−4

10−3

10−2

10−1

100

Nor

mal

ised

Sin

gula

r V

alue

s

N

Figure 3.1: The normalised singular values of matrixX for N0 = N ∈ 20 (− ·−), 40 (· · · ), 80 (−−−), for Example 1.

In what follows, we present numerical results obtained withN0 = N = 40. We

consider first the case of exact data, i.e. there is no noise inthe input data (3.4).

Figure 3.2(a) shows the analytical and numerical results ofr(t) from which it can be

Chapter 3. 69

seen that the numerical solution is very accurate. Figure 3.2(b) shows the numerical

solution ofu(0, t) in comparison with the analytical valueu(0, t) = et. The same

very good agreement between the numerical and analytical solutions is recorded. We

do not present the results for the flux atx = 1 since these were found to be equal to

the negative of the boundary temperature atx = 0. This is to be expected since from

(3.18) we haveu0(x) = u0(1 − x), f(x, t) = f(1 − x, t) and it can easily be verified

that from (3.8), using (3.3) withα = −1, it follows that q¯L

= −h¯0

. Furthermore, note

thatλ = 0, i.e. no regularisation, was found necessary forN0 = N = 40 and exact

input data (3.4).

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(b)

Figure 3.2: The analytical (—–) and numerical (−·−) results of (a)r(t) and (b)u(0, t)for exact data andλ = 0, for Example 1.

Next, we investigate the stability of the numerical solution with respect to noise

in the energy data (3.4), we add noise to the right-hand side of the oversdetermina-

tion condition (3.4),Eǫ, as generated in (3.14) with the standard deviation given by

σ =7ep

6. The contaminated data input withp = 1% is firstly inverted by using the

Gaussian elimination procedure as shown in Figure 3.3. Although from Figure 3.3(b)

the numerical solution foru(0, t) seems to remain stable and accurate, whereas Figure

3.3(a) shows that the numerical solution forr(t), with no regularisation, is unstable,

i.e. highly oscillatory and unbounded. This phenomenon is to be expected since the

inverse problem under investigation is ill-posed with the smallest normalised singular

Chapter 3. 70

value forN = 40 being ofO(103). Moreover, similar unstable results are obtained by

using the untruncated SVD or withλ = 0 in regularised solution (1.34), instead of the

Gaussian elimination method.

0 0.2 0.4 0.6 0.8 1−20

−15

−10

−5

0

5

10

15

20

25

30

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(b)

Figure 3.3: The analytical (—–) and numerical (−·−) results of (a)r(t) and (b)u(0, t)for p = 1% noisy data andλ = 0, for Example 1.

In order to avoid the over amplification of noise and maintainthe stability of the

results when the noise is presented, we employ the regularisation of either the TSVD

or the Tikhonov regularisation of orders zero, one and two. Furthermore, theL-curve

method and the discrepancy principle are investigated to indicate the truncation level

Nt for using TSVD, and choose the regularisation parameterλ for using the Tikhonov

regularisation. Firstly, we consider the BEM together withthe TSVD in the case of

p ∈ 1, 3, 5%. Initially, we have tried theL-curve criterion which should select the

truncated numberNt at the corner ofL-curve graph. Figure 3.4 displays theL-curves

for various percentages of noisep ∈ 1, 3, 5%. From this figure, especially on the nor-

mal scale one can see thatL-shaped curves appear and their corners indicate the level

of truncationNt in the following intervals9, . . . , 32, 12, . . . , 24 and10, . . . , 17for p ∈ 1, 3, 5%, respectively. Alternatively, one can choose the truncation level by

the more rigorous discrepancy principle which uses the knowledge of the level of noise

as in (3.16), i.e.ǫ = ‖y¯ǫ − y

¯‖ = ‖E

¯ǫ − E‖. Figure 3.5(a) represents the discrepancy

principle to selectNt for p ∈ 1, 3, 5%. These truncated numbersNt are obtained

Chapter 3. 71

within the intervals which have been recommended by theL-curve criterion of Figure

3.4. Besides, the numerical results forr(t) andu(0, t) are illustrated in Figures 3.5(b)

and 3.5(c), respectively, and the RMSE for these results aregiven in Table 3.1. From

Table 3.1 it can be remarked that the accuracy of the numerical results foru(0, t) im-

proves as the percentage of noise decreases; however, the convergence of the numerical

solution forr(t) towards the analytical solution is non-monotonic asp decreases (to

zero), see also Figure 3.5(b). Further, from Figure 3.5(c),it can be seen that the nu-

merical results foru(0, t) are stable, although by comparing with Figure 3.3(b) it can

be seen that forp = 1% noise the results with untruncation, i.e. no regularisation, are

slightly more accurate than the results withNt = 15. This is somewhat to be expected

since the pair of solutions(u(x, t), r(t)) of the inverse problem is stable in the temper-

atureu(x, t) component, but unstable in the heat sourcer(t) component. This latter

instability in r(t) which has already been illustrated in Figure 3.3(b), is alleviated in

Figure 3.5(b) which shows the numerical results with regularisation.

Alternatively for comparison, the zeroth-order Tikhonov regularisation (ZOTR) is

also utilised here. Initially, we have tried theL-curve plot of the residual‖Xr¯λ

−y¯ǫ‖ versus the norm of the solution‖r

¯λ‖, but we have found that anL-corner could

not be clearly identified. This is to be somewhat expected because theL-curve is a

heuristic method which is not always convergent [58]. Another optional method, the

more rigorous discrepancy principle which uses the knowledge of the level of noiseǫ

is recommended.

Figure 3.6(a) shows the discrepancy principle curves, for various percentages of

noisep ∈ 1, 3, 5% where the intersection of the residual curve with the horizontal

level noisy lineǫ yields the value ofλdis. The RMSE for the output solutions ofr(t)

andu(0, t) obtained with these values ofλdis are given in Table 3.1 and the numerical

results are illustrated in Figures 3.6(b) and 3.6(c). From this table it can be seen that the

ZOTR slightly outperforms the TSVD. As it can be seen from Figures 3.6(b) and 3.6(c)

the numerical results obtained by the ZOTR are stable and in good agreement with the

analytical solution. Although Figure 3.6(b) represents a significant improvement over

Chapter 3. 72

0.05 0.1 1 1010

12

14

16

18

20

22

24

26

28

Nt=1

Nt=15

Nt=20,...,32

Nt=19

Nt=39

‖Xr¯Nt

− y¯

ǫ‖

‖r ¯N

t‖

(a)

0 1 2 3 4 5 610

20

30

40

50

60

70

80

Nt=1

Nt=9,...,32

Nt=40

‖Xr¯Nt

− y¯

ǫ‖

‖r ¯N

t‖

(b)

100

10

20

30

40

50

Nt=1

Nt=10,11

Nt=9Nt=12,...,14

Nt=39

‖Xr¯Nt

− y¯

ǫ‖

‖r ¯N

t‖

Nt=15,...,24

(c)

0 1 2 3 4 5 60

20

40

60

80

100

120

140

160

Nt=1

Nt=40

‖Xr¯Nt

− y¯

ǫ‖

‖r ¯N

t‖

Nt=12,...,24

Nt=9

(d)

10−3

10−2

10−1

100

101

10

20

40

60

80

100

120

Nt=1

Nt=7,...,9

Nt=7

Nt=10,...,17

Nt=18,...,21

Nt=39

‖Xr¯Nt

− y¯

ǫ‖

‖r ¯N

t‖

(e)

0 1 2 3 4 510

20

30

40

50

60

70

80

90

100

110

Nt=1

Nt=7,...,9

Nt=10,...,17

Nt=18,...,21

Nt=40

‖Xr¯Nt

− y¯

ǫ‖

‖r ¯N

t‖

(f)

Figure 3.4: TheL-curve on (a), (c), (e) log-log scale and (b), (d), (f) normalscale,obtained using TSVD forp ∈ 1(top), 3(middle), 5(bottom)% noise, for Example1.

Chapter 3. 73

0 5 10 15 20 25 30 35 4010

−3

10−2

10−1

100

101

ε = 0.17Nt = 15

ε = 0.56Nt = 9

ε = 0.91Nt = 10

Nt

‖X

r ¯Nt−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

Figure 3.5: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), and (c)u(0, t) obtained using the TSVD forp ∈ 1(− ·−), 3(· · · ), 5(− − −)% noise at the truncation levelNt given in Table 3.1, for Ex-ample 1.

Figure 3.3(a) the numerical results are still oscillatory.In order to further improve on

the accuracy and stability of the numerical results of Figure 3.6(b), next we employ the

regularisation with the higher-order Tikhonov regularisation of orders one and two.

Figures 3.7 and 3.8 show the analogous numerical results to Figure 3.6, but for

the first- and second-order Tikhonov regularisation, denoted FOTR and SOTR, respec-

tively. The corresponding values of the regularisation parametersλdis chosen accord-

ing to the discrepancy principle are given in Table 3.1. By comparing Figures 3.6–3.8

and Table 3.1, it can be seen that the use of higher-order regularisation improves the

Chapter 3. 74

10−6

10−5

10−4

10−3

10−2

10−1

100

101

10−2

10−1

100

101

102

ε = 0.17λ = 8E−4

ε = 0.56λ = 4E−3

ε = 0.91λ = 5E−3

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

Figure 3.6: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), and (c)u(0, t) obtained using the ZOTR forp ∈ 1(− ·−), 3(· · · ), 5(−−−)% noise with the regularisation parametersλdis given in Table3.1, for Example 1.

accuracy and stability of the numerical results. This is to be expected since the analyt-

ical solution (3.19) to be retrieved is infinitely differentiable. However, if less smooth

sources are attempted to be retrieved, the use of higher-order regularisation does not

necessarily improve the accuracy of the numerical results,as can be seen from the next

example.

Chapter 3. 75

Table 3.1: The RMSE for the TSVD, ZOTR, FOTR, SOTR forp ∈ 0, 1, 3, 5% noise,for Example 1.

Regularisation p(%) parameterRMSE

r(t) u(0, t)

-0 0 8.1E-3 6.6E-41% 0 11.66 0.023

TSVD1% Nt=15 1.080 2.47E-23% Nt=9 1.443 6.03E-25% Nt=10 1.521 7.25E-2

ZOTR1% λdis=8E-4 0.878 2.40E-23% λdis=4E-3 1.324 6.22E-25% λdis=5E-3 1.280 9.90E-2

FOTR1% λdis=0.05 0.246 1.17E-23% λdis=0.45 0.435 3.14E-25% λdis=0.64 0.595 5.81E-2

SOTR1% λdis=5.2 0.113 6.86E-33% λdis=81 0.198 2.67E-25% λdis=324 0.473 5.48E-2

3.5.2 Example 2

The previous Example 1 involved retrieving a smooth source function given byr(t) =

e2t. In this example, we are considering the BEM combined only with the Tikhonov

regularisation as we found in the previous example that the ZOTR slightly outperforms

the TSVD. Consider a more severe discontinuous test function given by [63],

r(t) =

−1, t ∈ [0, 0.25),

1, t ∈ [0.25, 0.5),

−1, t ∈ [0.5, 0.75),

1, t ∈ [0.75, 1].

(3.20)

We also takeT = 1,α = −1, u0 = 0, andf = 1. Since the inverse problem (3.1)–(3.4)

does not have an analytical solution available for the temperatureu(x, t), the input en-

ergy data (3.4) is numerically simulated by solving first thedirect problem (3.1)–(3.2)

Chapter 3. 76

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

10−2

10−1

100

101

ε = 0.17λ = 0.05

ε = 0.56λ = 0.45

ε = 0.91λ = 0.64

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

Figure 3.7: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), and (c)u(0, t) obtained using the FOTR forp ∈ 1(− ·−), 3(· · · ), 5(−−−)% noise with the regularisation parametersλdis given in Table3.1, for Example 1.

with r given by (3.20). The numerical results forE(t) andu(0, t) are shown in Fig-

ures 3.9(a) and 3.9(b), respectively, for various numbers of boundary elements/cells

N = N0 ∈ 20, 40, 80. From Figure 3.9 it can be seen that the numerical solution is

convergent as the number of boundary elements increases. Also, there is little differ-

ence between the numerical results obtained with the various mesh sizes showing that

the independence of the mesh has been achieved. We can therefore take the numerical

results forE(t), simulated from the direct problem withN = N0 = 40 and shown in

Figure 3.9(a), as our input (3.4) in the inverse problem (3.1)–(3.4). Furthermore, in

Chapter 3. 77

10−3

10−2

10−1

100

101

102

103

104

105

106

0.1

0.2

0.4

0.6

0.8

1

ε = 0.17 λ = 5.2

ε = 0.56 λ = 81.9

ε = 0.91 λ = 324

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

t

u(0

,t)

(c)

Figure 3.8: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), and (c)u(0, t) obtained using the SOTR forp ∈ 1(− ·−), 3(· · · ), 5(−−−)% noise with the regularisation parametersλdis given in Table3.1, for Example 1.

order to avoid committing an inverse crime [32], in the inverse problem the number of

cells is takenN0 = 30 (different than 40) while the number of boundary elements is

kept the sameN = 40.

First, Figures 3.10(a) and 3.10(b) show the numerical results for r(t) andu(0, t),

respectively, for no noise in the input data (3.4) obtained with no regularisation, i.e.

λ = 0. The exact solution (3.20) forr(t) is also included, and the numerical solution

for u(0, t) from Figure 3.9(b) withN = N0 = 40 is also referred to as ‘analytical’.

From Figure 3.10 it can be seen that the agreement between thenumerical and the

Chapter 3. 78

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

E(t

)

(a)

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

u(0

,t)

(b)

Figure 3.9: The numerical results of (a)E(t) and (b)u(0, t) obtained by solving thedirect problem withN0 = N ∈ 20 (− · −), 40 (· · · ), 80 (−−−), for Example 2.

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

u(0

,t)

(b)

Figure 3.10: The analytical (—–) and numerical(− · −) results of (a)r(t) and (b)u(0, t) for exact data andλ = 0, for Example 2.

analytical solutions is excellent. Next we add noise to the input data (3.4) numerically

simulated in Figure 3.9(a). This is generated as in (3.14) with the standard deviation

given byσ = 0.19p. If λ = 0, the numerically retrieved results forr(t) were found

highly unbounded and oscillatory and therefore, they are not presented. The numerical

results for the discrepancy principle,r(t) andu(0, t) for various amounts of noise and

regularisation of various orders zero, one, and two are shown in Figures 3.11–3.13, re-

spectively, and Table 3.2. The use of higher-order regularisation imposes higher-order

Chapter 3. 79

Table 3.2: The RMSE for the ZOTR, FOTR, SOTR forp ∈ 1, 3, 5% noise andN = 40,N0 = 30, for Example 2.

Regularisation p(%)parameter RMSEλdis r(t) u(0, t)

ZOTR1% 8.7E-5 0.195 1.90E-33% 3.3E-4 0.284 4.66E-35% 4.3E-4 0.287 5.80E-3

FOTR1% 9.9E-5 0.211 1.98E-33% 8.1E-4 0.289 4.38E-35% 7.8E-4 0.280 4.69E-3

SOTR1% 7.7E-5 0.223 2.06E-33% 9.2E-4 0.292 4.48E-35% 7.3E-4 0.279 4.48E-3

smoothness of the desired output hence, since the solution (3.20) is discontinuous,

more accurate results are obtained with the ZOTR, whilst theFOTR and the SOTR,

although they achieve stability, they slightly oversmooththe retrieved solution.

3.6 Conclusions

In this chapter, the inverse problem of finding the time-dependent heat source together

with the temperature of heat equation under non-local boundary and integral overdeter-

mination conditions has been investigated. The general ill-posed problem, a numerical

method based on the BEM combined with either the TSVD or the Tikhonov regularisa-

tion has been proposed. The TSVD has been truncated at the optimal truncation level

given by theL-curve criterion and the discrepancy principle. Whereas the discrep-

ancy principle for choosing the regularisation parameter with three orders of Tikhonov

regularisation have also been employed. The numerical results were found to be ac-

curate and stable. These features of the numerical solutionincrease with decreasing

the amount of noise included in the input data and with increasing the order of reg-

ularisation for smooth sources. However, as expected, non-smooth sources are more

accurately reconstructed by lower-order regularisation.

Chapter 3. 80

10−6

10−5

10−4

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

ε = 0.011λ = 8.7E−5

ε = 0.037λ = 3.3E−4

ε = 0.059λ = 4.3E−4

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

u(0

,t)

(c)

Figure 3.11: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), and (c)u(0, t) obtained using the ZOTR forp ∈ 1(− ·−), 3(· · · ), 5(−−−)% noise with the regularisation parametersλdis given in Table3.2, for Example 2.

We have studied the nonlocal boundary and integral overdetermination condition

for the inverse source problem for the heat equation. In the next chapter, we will subject

these conditions to a coefficient identification problem forthe bioheat equation.

Chapter 3. 81

10−6

10−5

10−4

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

ε = 0.011λ = 9.8E−5

ε = 0.037λ = 8.1E−4

ε = 0.059λ = 7.8E−4

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

u(0

,t)

(c)

Figure 3.12: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), and (c)u(0, t) obtained using the FOTR forp ∈ 1(− ·−), 3(· · · ), 5(−−−)% noise with the regularisation parametersλdis given in Table3.2, for Example 2.

Chapter 3. 82

10−6

10−5

10−4

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

ε = 0.011λ = 7.7E−5

ε = 0.037λ = 9.2E−4

ε = 0.059λ = 7.3E−4

λ

‖X

r ¯λ−

y ¯ǫ‖

(a)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

t

r(t

)

(b)

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

u(0

,t)

(c)

Figure 3.13: (a) The discrepancy principle curve, and the analytical (—–) and nu-merical results of (b)r(t), and (c)u(0, t) obtained using the SOTR forp ∈ 1(− ·−), 3(· · · ), 5(−−−)% noise with the regularisation parametersλdis given in Table3.2, for Example 2.

Chapter 4


Coefficient in the Bioheat Equation

4.1 Introduction

The bioheat equation establishes a mathematical connection between the tissue tem-

perature and the arterial blood perfusion which are the dominant components in human

physiology, see [55]. It involves a blood perfusion coefficient whose determination is

of much interest, [53].

In the previous chapter, we have considered the identification of the time-dependent

heat sourcer(t) and the temperatureu(x, t) in the heat equation (3.1) with nonlocal

boundary and integral conditions. In this chapter, we aim tofind the time-dependent

blood perfusion coefficient functionP (t) and the temperature of the tissueu(x, t) en-

tering the bioheat equation (4.1) below. The initial (3.2),nonlocal boundary (3.3) and

integral (3.4) conditions are the same as in the previous chapter. We mention that time-

dependent coefficient identification problems with nonlocal boundary and/or integral

overdetermination conditions have recently attracted revitalising interest, e.g. the re-

construction of a time-dependent diffusivity [27], a bloodperfusion coefficient [23], or

a heat source [18, 24].

83

Chapter 4. 84

The inverse problem investigated in this chapter has already been proved to be

uniquely solvable in [33], but no numerical reconstructionhas been attempted. There-

fore, the purpose of this chapter is to devise a numerical stable method for obtaining

the solution of the inverse problem.


Let us consider the inverse problem consisting of finding thetime-dependent blood per-

fusion coefficient functionP (t) ∈ C[0, T ] and the temperature of the tissueu(x, t) ∈C2,1(DT )∩C1,0(DT ) satisfying the one-dimensional time-dependent bioheat equation,

[33, 54],

ut(x, t) = uxx(x, t)− P (t)u(x, t) + f(x, t), (x, t) ∈ DT , (4.1)

wheref is a known heat source term andL = 1 in the definition ofDT in (1.1), subject

to the following initial, boundary and overdetermination conditions:

u(x, 0) = u0(x), x ∈ [0, 1], (4.2)

u(0, t) = u(1, t), ux(0, t) + αu(0, t) = 0, t ∈ [0, T ], (4.3)∫ 1

0

u(x, t)dx = E(t), t ∈ [0, T ], (4.4)

where the functionu0 is given and it denotes the initial temperature,α is a given

constant heat transfer coefficient, andE represents the mass or energy of the system.

Note that the nonlocal periodic boundary condition (4.3) isencountered in biological

applications, [41], whilst the mass/energy specification (4.4) models processes related

to particle diffusion in turbulent plasma, [22], or heat conduction, [10]. The physical

constraint that the blood perfusionP (t) is positive can also be imposed, [37].

Note that the caseα = 0 has been dealt with in [24]. Herein, we consider the case

α 6= 0 whose unique solvability and local continuous dependence of the solution upon

Chapter 4. 85

the data of the inverse problem (4.1)–(4.4) have been established in [33]. Moreover,

the continuous dependence of the solution upon the data alsowas established, [21].

Consider now the following transformation, [11],

v(x, t) = r(t)u(x, t), r(t) = exp

(∫ t

0

P (τ) dτ

)

. (4.5)

Then the inverse problem (4.1)–(4.3) becomes

vt = vxx + r(t)f(x, t), (x, t) ∈ DT , (4.6)

v(x, 0) = u0(x), x ∈ [0, t], (4.7)

v(0, t) = v(1, t), vx(0, t) + αv(0, t) = 0, t ∈ [0, T ], (4.8)

with the transformed integral condition

∫ 1

0

v(x, t)dx = E(t)r(t), t ∈ [0, T ]. (4.9)

We also have thatr ∈ C1[0, T ], r(0) = 1, r(t) > 0, for t ∈ [0, T ]. Solving the inverse

problem (4.6)–(4.9) for the solution pair(r(t), v(x, t)) yields afterwards the solution

pair (P (t), u(x, t)) for the inverse problem (4.1)–(4.4) as given by

P (t) =r′(t)

r(t)and u(x, t) =

v(x, t)

r(t), (x, t) ∈ DT . (4.10)

From equation (4.10) one can observe that the ill-posednessof the inverse problem

consists of the numerical differentiation of the noisy function r(t) which would need

regularisation.


In this section, we apply the BEM to the one-dimensional inverse problem (4.6)–(4.9),

in order to approximate the solution(r(t), v(x, t)) which in turn, via (4.10), leads to

Chapter 4. 86

the original solution(P (t), u(x, t)) of the inverse problem (4.1)–(4.4). Utilising the

BEM is classical with the use of the fundamental solution forthe heat equation and

Green’s identities. As introduced in Section 1.3, we then obtain the boundary integral

equation

η(x)v(x, t) =

∫ t

0

[

G(x, t, ξ, τ)∂v

∂n(ξ)(ξ, τ)− v(ξ, τ)

∂G

∂n(ξ)(x, t, ξ, τ)

]

ξ∈0,1

dτ

+

∫ 1

0

G(x, t, y, 0)v(y, 0) dy+

∫ 1

0

∫ T

0

G(x, t, y, τ)r(τ)f(y, τ) dτdy,

(x, t) ∈ [0, 1]× (0, T ]. (4.11)

Using the same discretisation as in Chapters 1–3, then the integral equation above can

be approximate as

η(x)v(x, t) =

N∑

j=1

[A0j(x, t)q0j + ALj(x, t)qLj − B0j(x, t)h0j − BLj(x, t)hLj ]

+

N0∑

k=1

Ck(x, t)u0,k +N∑

j=1

Dj(x, t)rj . (4.12)

By applying the boundary condition (4.3), we obtain the system of2N linear equations,

the same as in (3.9),

(

A− 1

α(B +B∗)

)

q¯+ Cu

¯0+Dr

¯= 0

¯. (4.13)

The transformed integral condition (4.9) can also be expressed, via the midpoint nu-

merical integral approximation, as

1

N0

N0∑

k=1

v(xk, ti),=

∫ 1

0

v(x, ti) dx = E(ti)r(ti) = eiri, i = 1, N. (4.14)

Chapter 4. 87

By using the integral equation (4.12), via (4.13), and (4.14) as the same in Chapter 3

yields

1

N0

N0∑

k=1

[(

A(1)k − 1

α(B

(1)k +B

(1)∗

k )

)

q¯+ C

(1)k u

¯0+D

(1)k r

¯

]

= Er¯, (4.15)

whereE = diag(e1, eN ). Eliminating q¯

from (4.13) and (4.15) yields a linear system

of N equations

Xr¯= y

¯, (4.16)

with N unknowns, where

X =1

N0

N0∑

k=1

[

−(

A(1)k − 1

α(B

(1)k +B

(1)k

∗)

)(

A− 1

α(B +B∗)

)−1

D +D(1)k

]

− E,

y¯=

1

N0

N0∑

k=1

[

(

A(1)k − 1

α(B

(1)k +B

(1)k

∗)

)(

A− 1

α(B +B∗)

)−1

C − C(1)k

]

u¯0.

As we have mention before, this inverse problem is ill-posed, then we need to

employ the regularisation in order to retrieve the stability of the solution which will be

present in the next section.

4.4 Regularisation

In practical measurements, the data (4.4) is usually contaminated with noise. In order

to model this, we perturb (4.4) with random noise as defined in(3.14), i.e. E¯ǫ =

E¯+ ǫ. Then, from the contamination, it means that the left-hand side matrixX is

contaminated with noise, denoted byXǫ, where

ǫ ≈ ‖Xǫ −X‖. (4.17)

The norm of the matrix above is defined as the square root of thesum of squares of all

its elements. Hence, instead of (4.16) we have to solve the following linear system of

Chapter 4. 88

equations:

Xǫr¯= y

¯. (4.18)

When the noise is presented, we employ the Tikhonov regularisation, as described

in Subsection 1.6.2, yielding the regularised solution

r¯λ

=(

(Xǫ)TXǫ + λRTR)−1

(Xǫ)Ty¯. (4.19)

whereλ ≥ 0 is a regularisation parameter to be prescribed andR is a second-order

derivative regularisation matrix given by, [57],

R =1

(T/N)2

1 −2 1 0 0 .

0 1 −2 1 0 .

0 0 1 −2 1 .

. . . . . .

. (4.20)

Note that we have kept the multiple of1

(T/N)2to the regularisation matrix which

is different from the regularisation matrix introduced in Subsection 1.6.2, this is in

order to keep the scaling technique of the computation. In (4.19), the regularisation

parameter can be selected according to the GCV criterion which choosesλ > 0 as the

minimum of the GCV function, see e.g. [63] and (1.35),

GCV (λ) =‖Xǫr

¯λ− y

¯‖2

[trace(I −Xǫ((Xǫ)TXǫ + λRTR)−1(Xǫ)T)]2. (4.21)

The solution of the original inverse problem (4.1)–(4.4) can be obtained by substi-

tuting all approximate solutionsv, r, andr′ into (4.10). In order to obtain the solution

P (t), we also need to find the derivative functionr′(t) which can be approximated

using finite differences as

r′(t1) =rλ(t1)− 1

T/(2N), r′(ti) =

rλ(ti)− rλ(ti−1)

T/N, i = 2, N. (4.22)

Chapter 4. 89

In the next section, we will test numerical examples in orderto illustrate the accu-

racy and stability of the BEM combined with the regularisation technique.


This section presents two benchmark test examples in order to test the accuracy and

stability of the BEM numerical procedure introduced earlier. The RMSE defined in

(2.49) is also used here to evaluate the accuracy of the numerical results.

4.5.1 Example 1

We consider a benchmark test example with the inputT = 1, α = −1 and

u(x, 0) = u0(x) = 1 + x− x2, f(x, t) = (3 + x− x2)e−t,∫ 1

0

u(x, t) dx = E(t) = 7e−t/6,(4.23)

Then the analytical solution of the problem (4.1)–(4.4) is given by

u(x, t) = (1 + x− x2)e−t, P (t) = 2, (4.24)

whilst the analytical solution for the transformed problem(4.6)–(4.9) is

v(x, t) = (1 + x− x2)et, r(t) = e2t, (4.25)

In this example, we present the numerical results obtained with a BEM mesh ofN =

N0 = 40.

We start first with the case of exact data, i.e.p = 0. The numerical results for the

unknownsr(t), u(0, t), r′(t), andP (t) obtained using the straightforward inversion

r¯= X−1y

¯, i.e. without regularisationλ = 0 in (4.19), are compared with their cor-

responding analytic valuese2t, e−t, 2e2t, and 2, in Figures 4.1(a)–4.1(d), respectively.

Chapter 4. 90

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 12

4

6

8

10

12

14

16

t

r′(t

)

(c)

0 0.2 0.4 0.6 0.8 11.94

1.95

1.96

1.97

1.98

1.99

2

2.01

2.02

2.03

t

P(t

)

(d)

Figure 4.1: The analytical (—–) and numerical(− · −) results of (a)r(t), (b) u(0, t),(c) r′(t), and (d)P (t) obtained using no regularisation, i.e.λ = 0, for exact data, forExample 1.

From Figure 4.1 it can be seen that all the quantities of interest are accurate.

Next we investigate the stability of the numerical solutionwith respect to some

p = 1% noise included in the input energy dataE, as mentioned in Section 4.4. The

corresponding numerical results to Figure 4.1 (for exact data) are presented in Fig-

ure 4.2 (for noisy data). In Figures 4.2(a) and 4.2(b) the numerical results obtained

for r(t) andu(0, t), respectively, are relatively accurate. However, the numerical re-

sults obtained forr′(t) andP (t) = r′(t)/r(t) shown in Figures 4.2(c) and 4.2(d),

respectively, are highly unstable. This is because the differentiation of the noisy func-

tion r(t), shown in Figure 4.2(a) with dashed line, using the finite differences (4.22)

Chapter 4. 91

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 1−5

0

5

10

15

20

25

t

r′(t

)

(c)

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

t

P(t

)

(d)

Figure 4.2: The analytical (—–) and numerical(− · −) results of (a)r(t), (b) u(0, t),(c) r′(t), and (d)P (t) obtained using no regularisation, i.e.λ = 0, for p = 1% noise,for Example 1.

is an unstable procedure. In order to deal with this instability one can employ the

smoothing spline regularisation of [59], but this requiresthe knowledge of the discrep-

ancy between the analytical and numerical values ofr(t) in Figure 4.2(a), which is

not available if the analytical solution is not available. We shall elaborate in apply-

ing this technique later on for Example 2. Alternatively, weemploy the second-order

Tikhonov regularised solution (4.19) with the choice of theregularisation parameter

given by the minimum point of the GCV function (4.21). This isplotted in Figure 4.3

for p = 1% noise and the minimum yields the valueλGCV = 1.25 × 10−5. With the

value ofλGCV = 1.25 × 10−5, the solution (4.19) forr(t) is plotted in Figure 4.4(a).

Chapter 4. 92

By comparing with the previous unregularised solution shown in Figure 4.2(a), one

can now see that the obtained solution forr(t) is indeed smooth. Then the process

of numerical differentiation (4.22) is permitted and a stable approximation can be ob-

tained, as shown in Figures 4.4(c) and 4.4(d). There are someinaccuracies manifested

at the end pointst = 0 and 1, but this is commonly observed elsewhere when using

other stabilising techniques such as the mollification method or, the Tikhonov regular-

isation for the Fredholm integral equation of the first kind presented in detail in [56].

In Figure 4.4 we have also included for comparative purposesthe numerical results

obtained with the optimal value, in circle line( ) of the regularisation parameter

λopt = 1.05×10−6 selected by the trail and error. The RMSE forλGCV = 1.25×10−5

andλopt = 1.05 × 10−6 for the numerical results presented in Figure 4.4(d) in com-

parison with the analytical solution, are 0.324 and 0.219, respectively. Overall from

Figure 4.4 it can be seen that there are not much differences between the numerical

results obtained with the two values of the regularisation parameter. This confirms that

the GCV criterion performs well in choosing a suitable regularisation parameter for

obtaining a stable and accurate numerical solution.

10−7

10−6

10−5

10−4

6

6.5

7

7.5

8

8.5

9

9.5

10x 10

−5

λ = 1.25×10−5

λ

GC

V(λ

)

Figure 4.3: The GCV function (4.21), obtained using the second-order Tikhonov reg-ularisation forp = 1% noise, for Example 1.

Chapter 4. 93

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 10

5

10

15

t

r′(t

)

(c)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

t

P(t

)

(d)

Figure 4.4: The analytical (—–) and numerical results of (a)r(t), (b)u(0, t), (c) r′(t),and (d)P (t) obtained using the second-order Tikhonov regularisation with λGCV =1.25× 10−5 (− ·−) andλopt = 1.05× 10−6 ( ), for p = 1% noise, for Example 1.

4.5.2 Example 2

In the previous example, the BEM together with the second-order regularisation and

finite differences has been used to solve the inverse problemfor the bioheat equation

(4.1)–(4.4). In this example we present the BEM together with a smoothing spline

regularisation, to be utilised as another regularisation technique, for computing the

first-order derivative of a noisy function, see [59]. We consider the inverse problem

Chapter 4. 94

(4.1)–(4.4) with the inputL = T = 1, α = −1 and

u0(x) = x4(1− x)3, E(t) = 1280e−(t+t2/2),

f(x, t) = 6x2(x− 1)(7x2 − 8x+ 2)e−(t+t2/2).

(4.26)

Then the solution of the transformed inverse problem (4.6)–(4.9) is given by

v(x, t) = x4(1− x)3, r(t) = et+t2/2, (4.27)

whereas the solution of the original inverse bioheat conduction problem (4.1)–(4.4) is

u(x, t) = x4(1− x)3e−(t+t2/2), P (t) = 1 + t.

From Figure 4.2(a) of the previous example, we have noticed that the numerical

results for the perfusion coefficientP (t) are highly oscillatory and unbounded because

the numerical differentiation of a noisy function is an unstable procedure. For Example

1, we have used the second-order Tikhonov regularisation for solving the system of

equations (4.19) and this resulted in a smooth approximation for the functionr(t),

as shown in Figure 4.4(a). Alternatively, for Example 2 we investigate smoothing

a posteriorithe discrete noisy data r¯

obtained (without regularisation) like in Figure

4.2(a). This is applied as the smoothing spline technique of[59]. We are seeking

therefore a smooth functionr ∈ C1(R) with r′′ ∈ L2(R) which minimises the second-

order Tikhonov regularisation functional

IΛ(r) :=T

N

N∑

j=1

(r(tj)− rj)2 + Λ‖r′′‖2L2(R), (4.28)

whereΛ ≥ 0 is a regularisation parameter to be prescribed and

r¯= (rj)j=1,N = (Xǫ)−1y

¯, (4.29)

Chapter 4. 95

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3x 10

−5

t

u(0

,t)

(b)

0 0.2 0.4 0.6 0.8 1−10

−5

0

5

10

15

20

25

30

t

r′(t

)

(c)

0 0.2 0.4 0.6 0.8 1−2

−1

0

1

2

3

4

5

6

t

P(t

)

(d)

Figure 4.5: The analytical (—–) and numerical results of (a)r(t), (b) u(0, t)(t), (c)r′(t), and (d)P (t) obtained using no regularisation in (4.31), i.e.Λ = 0, for exact data( ) and forp = 1% noisy data(− · −), for Example 2.

obtained from Figure 4.2(a). We further approximate the functionr using cubic splines

as

r(t) = d1 + d2t+

N∑

j=1

cj|t− tj |3, (4.30)

where the coefficients(cj)j=1,N and(dj)j=1,2 satisfy the conditions (4.31) and (4.32)

below. Inserting (4.30) into (4.28) and minimising the resulting expression with re-

spect to the coefficients(cj)j=1,N and(dj)j=1,2 yields the following system of(N +2)

Chapter 4. 96

equations with(N + 2) unknowns, [59],

rΛ(tj) + 12ΛNcj = rj , j = 1, N, (4.31)

N∑

j=1

cj =

N∑

j=1

cjtj = 0. (4.32)

By solving the system of equations (4.31) and (4.32) we obtain the coefficients(cj)j=1,N

and(dj)j=1,2, and therefore the expression for the smooth functionr, given by (4.30).

By differentiating this expression with respect tot we obtain the first-order derivative

r′(t) given by

r′(t) = d2 + 3

N∑

j=1

cj(t− tj)2sign(t− tj), (4.33)

where sign(·) is the signum function. This derivativer′(t) is presented asr′(t) in

(4.27). From (4.31), one can observe that ifΛ = 0 thenr0(ti) = r(ti) for i = 1, N .

In the case of exact data actually one can takeΛ = 0, but for noisy data takingΛ = 0

produces a highly unbounded and oscillatory solution forr′(t) andP (t), as shown in

Figures 4.5(c) and 4.5(d). For noisy data, reference [59] suggests thea priori choice

Λ =T

N

N∑

j=1

(rexact,j − rj)2, (4.34)

whererexact,j = rexact(tj) represents an analytical solutionr, given by (4.27), at time

tj for j = 1, N . Forp = 1% noise, we haveΛ ≈ 8 × 10−3. The numerical results for

r′(t) andP (t) obtained using the smoothing spline regularisation technique with this

choice ofΛ oversmoothes the exact solution, as shown in Figure 4.6 in dash-dot line

(− · −). However, the Morozov’s discrepancy principle [40] based on thea posteriori

choice ofΛ such that

T

N

N∑

j=1

(rΛ(tj)− rj)2 ≈ T

N

N∑

j=1

(rexact,j − rj)2, (4.35)

producesΛdis = 7 × 10−5 which yields more accurate approximations, as also shown

Chapter 4. 97

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

9

t

r′(t

)

(a)

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

t

P(t

)

(b)

Figure 4.6: The analytical (—–) and numerical results of (a)r′(t) and (b)P (t) obtainedusing the smoothing spline regularisation withΛdis = 7×10−5 () andΛ = 8×10−3

(− · −), as defined in (4.34), forp = 1% noise, for Example 2.

in Figure 4.6 in circle line( ).Finally, although not illustrated, it is reported that for higher amounts of noise the

numerical reconstructions are less stable.

4.6 Conclusions

The inverse problem of finding a time-dependent blood perfusion coefficient for the

bioheat equation with nonlocal boundary conditions and mass/energy specification has

been investigated. The inverse problem has been transformed to an inverse heat source

problem with an unknown term present in the integral over-determination condition.

The numerical discretisation was based on the BEM together with the Tikhonov reg-

ularisation and the GCV for the choice of the regularisationparameter. We have also

applied the smoothing spline technique for differentiating a noisy function witha pri-

ori anda posteriorichoices of the regularisation parameters. For a couple of typical

benchmark test examples, accurate and stable numerical solutions have been obtained.

Chapter 5. 98

Chapter 5


Heat Source with a Dynamic

Boundary Condition

5.1 Introduction

In the Chapters 3 and 4, we have considered inverse time-dependent source problems

for the heat equation with various types of conditions such as integral, local or nonlo-

cal. In the present chapter, we consider yet another reconstruction of a time-dependent

heat source with the integral over-determination measurement of the thermal energy

of the system and a new dynamic-type boundary condition. This model can be used

in heat transfer and diffusion processes with a time-dependent source parameter to be

determined. Also, in acoustic scattering or damage corrosion the new dynamic-type

boundary condition (5.4) below is also known as a generalised impedance boundary

condition, [4–7].

The well-posedness of the inverse problem studied in this chapter was established

in [19], and we aim to obtain the numerical solution by using the BEM together with a

regularisation method.

99

Chapter 5. 100

This chapter is organised as follows. In Section 5.2, the mathematical formula-

tion of the inverse problem is described. The numerical discretisation of the problem

based on the BEM is described in Section 5.3. Section 5.4 discusses numerical results

obtained for three of benchmark test examples and emphasises the importance of em-

ploying the Tikhonov regularisation with the choice of regularisation parameter based

on either the GCV criterion or the discrepancy principle, inorder to achieve a stable

numerical solution. Finally, Section 5.5 presents the conclusions of the chapter.


Consider the following initial-boundary value problem of finding the time-depending

heat sourcer(t) ∈ C([0, T ]) and the temperatureu(x, t) ∈ C2,1(DT ) ∩ C1,0(DT )

which satisfy the heat equation:

ut = uxx + r(t)f(x, t), (x, t) ∈ DT , (5.1)

whereL = 1 in the definition ofDT in (1.1), subject to the initial condition (1.7),

namely

u(x, 0) = u0(x), x ∈ [0, 1], (5.2)

and the boundary conditions

u(0, t) = 0, t ∈ (0, T ], (5.3)

auxx(1, t) + αux(1, t) + bu(1, t) = 0, t ∈ (0, T ], (5.4)

wheref andu0 are given functions anda, b, α are given numbers not simultaneously

equal to zero. The well-posedness of this direct problem wasestablished in [34].

Taking into account the equation (5.1) atx = 1, the boundary condition (5.4)

Chapter 5. 101

becomes

aut(1, t) + αux(1, t) + bu(1, t) = ar(t)f(1, t), t ∈ (0, T ]. (5.5)

In order to add further physics to the problem, we mention that the boundary condition

(5.5) is observed in the process of cooling of a thin solid barone end of which is placed

in contact with a fluid [36]. Another possible application ofsuch type of boundary

condition is announced in [9, p.79], as this boundary condition represents a boundary

reaction in diffusion of a chemical. We finally mention that we have also previously

encountered the dynamic boundary condition (5.5) when modelling a transient flow

pump experiment in a porous medium [39].

When the functionr(t) for t ∈ [0, T ] is unknown, the inverse problem formulates

as a problem of finding a pair of functions(r(t), u(x, t)) which satisfy the equation

(5.1), initial condition (5.2), the boundary conditions (5.3) and (5.4) (or (5.5)), and the

energy/mass overdetermination measurement

∫ 1

0

u(x, t) dx = E(t), t ∈ [0, T ]. (5.6)

This overdetermination condition is encountered in modelling applications related to

particle diffusion in turbulent plasma, as well as in heat conduction problems in which

the law of variationE(t) of the total energy of heat in a rod is given, [22].

If we let u(x, t) represent the temperature distribution, then the above-mentioned

inverse problem can be regarded as a source control problem.The source control

parameterr(t) needs to be determined from the measurement of the thermal energy

E(t).

DenoteΦ4n0[0, 1] := φ(x) ∈ C4[0, 1];φ(0) = φ′′(0) = 0, φ(1) = φ′(1) =

φ′′(1) = φ′′′(1) = 0,∫ 1

0φ(x)yn0

(x)dx = 0, where(yn)n≥0 denote the eigenfunctions

Chapter 5. 102

of the spectral problem

y′′(x) + µy(x) = 0, x ∈ [0, 1],

y(0) = 0,

(aµ− b)y(1) = αy′(1),

(5.7)

where its eigenvalueµn and eigenfunctionsyn(x), for n = 0, 1, 2, . . . , have the fol-

lowing asymptotic behavior:

√µn = πn +O

(

1

n

)

, yn(x) = sin(πnx) +O

(

1

n

)

,

for sufficiently largen. The following theorem proved in [19] established the existence

of a unique solution of the inverse problem (5.1)–(5.3), (5.5) and (5.6).

Theorem 5.2.1 Letaα > 0 and assume that the following conditions are satisfied:

(A1) u0(x) ∈ Φ4n0[0, 1];

(A2) E(t) ∈ C1[0, T ]; E(0) =∫ 1

0u0(x)dx;

(A3) f(x, t) ∈ C(DT ); f(·, t) ∈ Φ4n0[0, 1], ∀t ∈ [0, T ]; and

∫ 1

0f(x, t)dx 6= 0, ∀t ∈

[0, T ];

Then the inverse problem(5.1)–(5.3), (5.5)and(5.6)has a unique solution(r(t), u(x, t)) ∈C[0, T ]× (C2,1(DT ) ∩ C2,0(DT )). Moreover,u(x, t) ∈ C2,1(DT ).

Although the inverse problem is uniquely solvable, it is still ill-posed since small errors

in the input data (5.6) cause large errors in the input sourcer(t).

In the next section we will describes the discretisation of the inverse problem using

the BEM, whilst Section 5.4 will discuss the regularisationof the numerical solution.

Chapter 5. 103

5.3 Boundary element method (BEM)

In this section, we explain the numerical procedure for discretising the inverse problem

(5.1)–(5.3), (5.5) and (5.6) by using the BEM. As introducedin Section 1.3, the use

of BEM recasts the heat equation (5.1) in the boundary integral form (3.5). Using the

same discretisation as described in Chapters 2–4, and applying the boundary condition

(5.3), the boundary integral equation (3.5) becomes

η(x)u(x, t) =

N∑

j=1

[A0j(x, t)q0j + ALj(x, t)qlj −BLj(x, t)hLj ]

+

N0∑

k=1

Ck(x, t)u0,k +N∑

j=1

Dj(x, t)rj , (x, t) ∈ [0, 1]× (0, T ]. (5.8)

On applying the BEM and the boundary condition (5.3), we obtain the system of2N

linear equations,

A0q¯0

+ ALq¯L

− BLh¯L

+ Cu¯0

+Dr¯= 0

¯. (5.9)

In order to apply the boundary condition (5.5) we need to approximate the time-

derivativeut(1, t) by using finite differences. For this, we use theO(h2) finite differ-

ence formulae

ut(1, t1) =u(1, t2)/3 + u(1, t1)− 4u0(1)/3

h,

ut(1, t2) =5u(1, t2)/3− 3u(1, t1) + 4u0(1)/3

h,

ut(1, ti) =3u(1, ti)/2− 2u(1, ti−1) + u(1, ti−2)/2

h, i = 3, N,

where the step sizeh = T/N and ti = ti−1+ti2

as defined in Chapter 1. Applying

the expressions above into the boundary condition (5.5) yields the linear system ofN

Chapter 5. 104

equations as follows:

a

hu(1, t1) +

a

3hu(1, t2) + bu(1, t1) = ar(t1)f(1, t1) +

4a

3hu0(1)− αux(1, t1),

−3a

hu(1, t1) +

5a

3hu(1, t2) + bu(1, t2) = ar(t2)f(1, t2)−

4a

3hu0(1)− αux(1, t2),

a

2hu(1, ti−2)−

2a

hu(1, ti−1) +

3a

2hu(1, ti) + bu(1, ti) = ar(ti)f(1, ti)− αux(1, ti),

for i = 3, N . This system can be rewritten as

Sh¯L

= Fr¯+ u

¯0− αq

¯L, (5.10)

whereF = diag(af(1, t1), . . . , af(1, tN)), and

S =

a/h + b a/3h 0 .

−3a/h 5a/3h+ b 0 .

a/2h −2a/h 3a/2h+ b .

. . . .

0 a/2h −2a/h 3a/2h+ b

N×N

, u¯0

=

4au0(1)/3h

−4au0(1)/3h

0

.

0

N

.

Assumingα 6= 0, eliminating q¯L

can be done by applying the derived matrix form of

(5.10), i.e.

q¯L

=1

α(Fr

¯+ u

¯0− Sh

¯L) , (5.11)

into (5.9), this gives (5.9) and (5.10) results as

h¯L

q¯0

=

[(

1

αALS +BL

)

∣

∣

∣−A0

]−1(1

αALFr

¯+

1

αALu

¯0+ Cu

¯0+Dr

¯

)

, (5.12)

where the invertible matrix[

(

1αALS +BL

)

∣

∣

∣−A0

]

is a 2N × 2N matrix formed

with the 2N × N block matrices(

1αALS +BL

)

and−A0 separated by the vertical

line.

Next, we collocate the over-determination condition (5.6)by using the midpoint

Chapter 5. 105

numerical integration approximation as same as in (3.11). Then the expression (5.8) at

(xk, ti), can be rewritten as

1

N0

N0∑

k=1

[

A(1)0,kq

¯0+ A

(1)L,kq

¯L− B

(1)L,kh¯L

+ C(1)k u

¯0+D

(1)k r

¯

]

= E¯, (5.13)

where

A(1)0,k =

[

A0j(xk, ti)]

N×N, A

(1)L,k =

[

ALj(xk, ti)]

N×N, B

(1)L,k =

[

BLj(xk, ti)]

N×N,

C(1)k =

[

Cl(xk, ti)]

N×N0

, D(1)k =

[

Dj(xk, ti)]

N×N, k, l = 1, N0, i, j = 1, N.

When the boundary condition (5.3) is applied, the expression (5.13) can be, via (5.11),

rewritten as

1

N0

N0∑

k=1

[

A(1)0,kq

¯0+

1

αA

(1)L,k (Fr

¯+ u

¯0− Sh

¯L)− B

(1)L,kh¯L

+ C(1)k u

¯0+D

(1)k r

¯

]

= E¯. (5.14)

Finally, eliminating q¯0

and h¯L

between (5.12) and (5.14), the unknown discretised

source r¯

can be found by solving theN ×N linear system of equations

Xr¯= y

¯, (5.15)

where

X =1

N0

N0∑

k=1

[(

1

αA

(1)L,kS +B

(1)L,k

)

∣

∣

∣−A(1)

0,k

] [(

1

αALS +BL

)

∣

∣

∣−A0

]−1

×(

1

αALF +D

)

−(

1

αA

(1)L,kF +D

(1)k

)

,

y¯=

1

N0

N0∑

k=1

C(1)k u

¯0+

1

αA

(1)L,ku¯0

−[(

1

αA

(1)L,kS +B

(1)L,k

)

∣

∣

∣−A(1)

0,k

]

×[(

1

αALS +BL

)

∣

∣

∣−A0

]−1(

Cu¯0

+1

αALu

¯0

)

− E¯.

Chapter 5. 106

As we have mentioned previously the inverse problem under investigation is ill-

posed, and consequently, the system of equation (5.15) is ill-conditioned. In the next

section, we will discuss the regularisation of the numerical solution together with

choices of regularisation parameter based on either the GCVcriterion or the discrep-

ancy principle.


This section presents three benchmark test examples with smooth and non-smooth con-

tinuous source functions in order to test the accuracy of theBEM numerical procedure

introduced earlier in Section 5.3. In order to illustrate the accuracy of the numerical

results, the RMSE defined in (2.49) is also used here.

5.4.1 Example 1

In the first example, we consider the case of smooth continuous unknown source func-

tion, given by the analytical solution

u(x, t) = x2et, r(t) = et, (5.16)

for the inverse problem (5.1)–(5.3), (5.5) and (5.6) with the input dataT = 1, a = α =

1, b = −4,

u(x, 0) = u0(x) = x2, f(x, t) = x2 − 2, (5.17)

The direct problem (5.1)–(5.3) and (5.5), whenr(t) = et is known, is considered first

with N = N0 ∈ 20, 40, 80 obtained by (5.11), (5.12) and (5.13), and the RMSE

results are shown in Table 5.1. From this table it can be concluded that the BEM

numerical solution is convergent to the corresponding exact values

u(1, t) = et, ux(0, t) = 0, ux(1, t) = 2et, E(t) = et/3, (5.18)

Chapter 5. 107

for t ∈ [0, 1], as the number of boundary elements increases.

Table 5.1: The RMSE foru(1, t), ux(0, t), ux(1, t) andE(t) obtained using the BEMfor the direct problem withN = N0 ∈ 20, 40, 80, for Example 1.

N = N0RMSE

u(1, t) ux(0, t) ux(1, t) E(t)20 6.43E-3 2.79E-3 8.85E-3 2.65E-340 2.20E-3 9.68E-4 2.98E-3 9.07E-480 7.46E-4 3.32E-4 1.00E-3 3.08E-4

Next, we consider the inverse problem (5.1)–(5.3), (5.5) and (5.6) and we use the

BEM with N = N0 = 40 for solving the resulting system of equations (5.15). Figure

5.1 displays the analytical and numerical results ofr(t), u(1, t), ux(0, t), andux(1, t)

and very good agreement can be observed.

In practice, the contamination of measured data by unplanned error is unavoidable.

Thus we add noise to the input energy dataE(t) in (5.6) in order to test the stability of

the solution. Here, the perturbed input data E¯ǫ is defined as same as described in (3.14),

with the standard deviationσ =ep

3andp ∈ 1, 3, 5%. This perturbation means that

the known right-hand side vector y¯

of the linear system (5.15) is contaminated with

noise, denoted as y¯ǫ. Then, when noise is present, we have to solve the following

system of linear equations instead of (5.15):

Xr¯= y

¯ǫ. (5.19)

Initially, we have tried to solve the above disturbed system(5.19) withp = 1% noise

in the input data (5.6) by using the straightforward inversion of (5.19), i.e. r¯= X−1y

¯ǫ,

illustrated in Figure 5.2. From this figure it can be seen thatthe numerical solutions for

r(t), ux(0, t) andux(1, t) shown by the dash-dot line (− · −) are unstable. However,

the result foru(1, t) seems to remain stable.

To overcome this instability, we employ the Tikhonov regularisation method as

described in (1.34) with a second-order differential regularisation matrix in (4.20). As

happened previously with some of our investigations in Chapters 2 and 3, we report that

Chapter 5. 108

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 1−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1x 10

−3

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 12

2.5

3

3.5

4

4.5

5

5.5

t

ux(1

,t)

(d)

Figure 5.1: The analytical (—–) and numerical results(− · −) of (a) r(t), (b) u(1, t),(c) ux(0, t), and (d)ux(1, t) for exact data, for Example 1.

the second-order Tikhonov regularisation has produced more accurate results than the

zeroth- or first-order regularisation and therefore, only the numerical results obtained

using the former regularisation are illustrated in this section.

A popular method for choosing the regularisation parameteris the GCV criterion

which is based on the minimisation technique, as we have detailed in Section 1.6. For

p = 1% noise, this minimisation yields the minimum point of the GCVfunction (1.35)

occurring atλGCV =4.3E-6. Then the numerical results obtained using the Tikhonov

regularisation with this value ofλGCV , illustrated by circles( ) in Figure 5.2, show

that accurate and stable numerical solutions are achieved.

Next, we increase top = 3% and5% the percentage of noise. Figure 5.3 presents

Chapter 5. 109

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

6

8

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11.5

2

2.5

3

3.5

4

4.5

5

5.5

t

ux(1

,t)

(d)

Figure 5.2: The analytical (—–) and numerical results of (a)r(t), (b) u(1, t), (c)ux(0, t), and (d)ux(1, t) obtained using the straightforward inversion(− · −) withno regularisation, and the second-order Tikhonov regularisation( ) with the regu-larisation parameterλ=4.3E-6 suggested by the GCV method, forp = 1% noise, forExample 1.

the analytical and numerical results obtained using the second-order Tikhonov reg-

ularisation with the regularisation parameter suggested by the GCV method, namely

λGCV =7.4E-6 forp = 3%, andλGCV =2.7E-5 forp = 5%. From this figure one can ob-

serve that stable and accurate results forr(t), u(1, t), ux(0, t) andux(1, t) with p = 3%

noise are attained, whereas the numerical results forp = 5% noisy input are rather in-

accurate, but they remain stable. For completeness, the RMSE errors are displayed in

Table 5.2.

Chapter 5. 110

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 12

2.5

3

3.5

4

4.5

5

5.5

t

ux(1

,t)

(d)

Figure 5.3: The analytical (—–) and numerical results of (a)r(t), (b) u(1, t), (c)ux(0, t), and (d)ux(1, t) obtained using the second-order Tikhonov regularisation withthe regularisation parameter suggested by the GCV method, for p = 3% (· · ·) andp = 5% (−−−), for Example 1.

5.4.2 Example 2

The previous example possessed an analytical solution being explicitly available; how-

ever the source functionf(x, t) chosen did not satisfy the condition in (A3) of Theorem

5.2.1 thatf ∈ Φ4n0[0, 1]. Therefore, in this subsection we aim to construct an example

for which the conditions of existence and uniqueness of solution of Theorem 5.2.1 are

satisfied. We chooseT = 1, a = α = 1, b = 0 andu0(x) = 0.

In the casea = α = 1, b = 0 the problem (5.7) has the eigenvaluesµn = ν2n,

whereνn are the positive roots of the transcendental equationν sin(ν) = cos(ν). The

Chapter 5. 111

Table 5.2: The regularisation parametersλ and the RMSE forr(t), u(1, t), ux(0, t) andux(1, t), obtained using the BEM withN = N0 = 40 combined with the second-orderTikhonov regularisation forp ∈ 0, 1, 3, 5% noise, for Example 1.

pparameter RMSE

λ r(t) u(1, t) ux(0, t) ux(1, t)0 0 4.16E-3 2.47E-4 1.20E-3 8.85E-4

1% 0 2.70 1.72E-2 1.12E-1 2.64E-11% λGCV =4.3E-6 1.73E-2 2.57E-3 8.92E-3 5.47E-33% 0 5.21 4.13E-2 3.51E-1 5.02E-13% λGCV =7.4E-6 3.32E-2 9.73E-3 1.97E-2 2.25E-25% 0 4.74 5.51E-2 4.64E-1 4.54E-15% λGCV =2.7E-5 1.95E-1 4.63E-2 1.29E-1 9.79E-2

corresponding eigenfunctions areyn(x) = sin(νnx). The first eigenvalue is given by

ν0 =√µ0 = 0.860333. Then choosing

f(x, t) = x3(1− x)4(β1x+ β2), (5.20)

we can determine the constantsβ1 andβ2 such thatf ∈ Φ40[0, 1] (choosingn0 = 0 for

simplicity), as required by the condition (A3) of Theorem 5.2.1, i.e.f(x, t) ∈ Φ4n0[0, 1],

∀t ∈ [0, t]. This imposes

0 =

∫ 1

0

f(x, t) sin(ν0x) dx =

∫ 1

0

x3(1− x)4(β1x+ β2) sin(ν0x) dx.

After some calculus, choosingβ2 = −1 it follows thatβ1 ≈ 2.011. With these values

of β1 andβ2 we also satisfy that∫ 1

0f(x, t) dx = −0.00037 is non-zero, as required by

condition (A3). We aim to retrieve a non-smooth source function given by

r(t) =

∣

∣

∣

∣

t− 1

2

∣

∣

∣

∣

, t ∈ [0, t]. (5.21)

In this case, the analytical solution of the direct problem for the temperatureu(x, t)

is not available. Thus the energyE(t) is not available either. In such a situation, we

simulate the data (5.6) numerically by solving first the direct problem (5.1)–(5.3) and

Chapter 5. 112

(5.5) with r known and given by (5.21). The numerical solutions foru(1, t), ux(0, t),

ux(1, t) andE(t) obtained using the BEM withN = N0 ∈ 20, 40, 80 are shown

in Figure 5.4. From this figure it can be seen that convergent numerical solutions are

obtained.

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

−6

t

u(1

,t)

(a)

0 0.2 0.4 0.6 0.8 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10

−4

t

ux(0

,t)

(b)

0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1x 10

−5

t

ux(1

,t)

(c)

0 0.2 0.4 0.6 0.8 1−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10

−5

t

E(t

)

(d)

Figure 5.4: The numerical results of (a)u(1, t), (b) ux(0, t), (c) ux(1, t), and (d)E(t)obtained by solving the direct problem withN = N0 ∈ 20(), 40(···), 80(−−−),for Example 2.

To investigate the inverse problem (5.1)–(5.3), (5.5) and (5.6), we use the numerical

results forE(t) in Figure 5.4(d) obtained using the BEM withN = N0 = 40, as the

input data (5.6). In order to avoid committing an inverse crime we keepN = 40, but

we use a differentN0, sayN0 = 30, than 40 which was used in the direct problem

simulation. Figure 5.5 shows the numerical results obtained without regularisation,

Chapter 5. 113

i.e. λ = 0, for p = 0 (exact) andp = 1% (noisy) data. Remark that from Figure 5.4(d),

the standard deviation for generating noise is given byσ = 1.2 × 10−5p. From Figure

5.5 it can be seen that, for exact data, the straightforward inversion of (5.15) produces

very accurate results. However, when noise is introduced into the measured data E¯ǫ,

here we are solving the linear system of equationsXr¯= y

¯ǫ instead ofXr

¯= y

¯, the

numerical retrievals of especiallyr(t) andux(1, t) become highly oscillatory unstable.

In order to retrieve the stability, as in Example 1, the second-order Tikhonov reg-

ularisation with the GCV criterion are employed and the numerically obtained results

are shown in Figure 5.6. The numerical results from the direct problem presented in

Figures 5.7(a)–5.7(c) are used to compare in Figures 5.6(b)–5.6(d) the numerical re-

sults foru(1, t), ux(0, t), andux(1, t), respectively, of the inverse problem. Whereas

the numerical solution forr(t) of the inverse problem is compared with the analytical

solution (5.21) in Figure 5.6(a). From Figure 5.6 it can be seen that stable and accurate

numerical solutions are obtained. For completeness, the RMSE errors and the GCV

values forλ are displayed in Table 5.3.

Table 5.3: The regularisation parametersλ and the RMSE forr(t), u(1, t), ux(0, t)andux(1, t), obtained using the BEM withN = 40 andN0 = 30 combined with thesecond-order Tikhonov regularisation forp ∈ 0, 1, 3, 5% noise, for Example 2.

p(%)parameter RMSE


1% 0 9.98E-2 2.71E-8 9.55E-6 3.55E-61% λGCV =3.2E-16 5.47E-3 1.62E-8 1.28E-6 4.12E-73% 0 3.63E-1 1.05E-7 3.39E-5 1.27E-53% λGCV =1.1E-15 1.37E-2 5.96E-8 3.56E-6 9.37E-75% 0 5.03E-1 1.70E-7 4.85E-5 1.80E-55% λGCV =9.0E-16 2.17E-2 9.88E-8 5.68E-6 1.21E-6

If one would like to make a fair comparison between the accuracy of the numerical

results obtained for Examples 1 and 2, the RMSE values presented in Tables 5.2 and

5.3 should be divided by the maximum absolute values of the corresponding quantities

involved. For example, if we divide the columns of RMSE values for r(t) in Tables

Chapter 5. 114

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

3.5x 10

−6

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0x 10

−4

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1

2x 10

−5

t

ux(1

,t)

(d)

Figure 5.5: The analytical solution (5.21) and the direct problem numerical solutionfrom Figures 5.7(a)–5.7(c) (—–) and numerical results of (a) r(t), (b) u(1, t), (c)ux(0, t), and (d)ux(1, t), with no regularisation, for exact data( ) and noisy datap = 1% (− · −), for Example 2.

5.2 and 5.3 bye (maximum value ofr(t) in (5.16)) and0.5 (maximum value ofr(t) in

(5.21)), respectively, then the relative errors forr(t) in Example 1 are actually lower

than those in Example 2, as expected from the regularity of these solution.

Finally, although not illustrated, it is reported that for both Examples 1 and 2 we

have experienced with other values ofλ close to the optimal ones but there was not

much difference obtained in comparison with the numerical results of Figures 5.2, 5.3

and 5.6. This confirms that the GCV criterion performs well inchoosing a suitable

regularisation parameter for obtaining a stable and accurate numerical solution.

Chapter 5. 115

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

3.5x 10

−6

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0x 10

−4

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1

−3

−2

−1

0

1

x 10−5

t

ux(1

,t)

(d)

Figure 5.6: The analytical solution (5.21) and the direct problem numerical solu-tions from Figures 5.7(a)–5.7(c) (—–), and the numerical results of (a)r(t), (b)u(1, t), (c) ux(0, t), and (d)ux(1, t) obtained using the second-order Tikhonov reg-ularisation with the regularisation parameters suggestedby GCV method, forp ∈1( ), 3(· · · ), 5(−−−)% noise, for Example 2.

5.4.3 Example 3

In previous examples, we have use the BEM together with the second-order Tikhonov

regularisation with the value of the regularisation parameter suggested by the GCV

method, on both cases for identification of the smooth and non-smooth source func-

tions in Example 1 and 2, respectively. In this example, we consider yet another case

Chapter 5. 116

example of non-smooth source function given by

r(t) =

∣

∣

∣

∣

t− 1

2

∣

∣

∣

∣

,

with the input dataT = 1, a = α = 1, b = 0,

u0(x) = 0, f(x, t) = 1. (5.22)

We remark that the condition in(A3) of Theorem 5.2.1 does not hold.

In this case, as with Example 2 that the analytical solution of the direct problem

for the temperatureu(x, t) is not available, thus the energyE(t) is not available either.

Therefore as we have done in previous example, the direct problem has been solved

first with the BEM andN = N0 ∈ 20, 40, 80, as illustrated in Figure 5.7. From

this figure it can be seen that rapidly convergent numerical solutions are obtained. The

numerical results obtained using the BEM withN = N0 = 40 are kept as an input data

for E(t) and the reference (analytical) solution foru(1, t), ux(0, t) andux(1, t).

In what follows, we present numerical results obtained withN = 40 andN0 = 30

instead of 40, in order to avoid committing an inverse crime.The energy dataE(t)

obtained from the numerical result in Figure 5.7(d) is perturbed, as described in (3.14)

with the standard deviationσ = 0.15p. Figure 5.8 shows that the numerical results

obtained by the straightforward inversion of (5.15), i.e. r¯= X−1y

¯, without regularisa-

tion, for p = 0 (exact) andp = 1% (noisy) data. For exact data, the same very good

agreement between the analytical and numerical solutions is recorded. Whereas for

the noisy data, the numerical solutions forr(t) andux(1, t) becomes highly oscilla-

tory and unbounded. This is somewhat to be expected since theinverse problem under

investigation is ill-posed.

We retrieve the stability, as previous, by the second-orderTikhonov regularisation

with the proper choice of the regularisation parameterλ. Firstly, we have used the rig-

orous discrepancy principle, as introduced in Section 1.6.3, for p ∈ 1, 3, 5% which

yieldsλdis ∈ 5.7E-8, 2.0E-6, 7.0E-6, respectively. The results are obtained as shown

Chapter 5. 117

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t

u(1

,t)

(a)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

ux(0

,t)

(b)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

t

ux(1

,t)

(c)

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

t

E(t

)

(d)

Figure 5.7: The numerical results of (a)u(1, t), (b) ux(0, t), (c) ux(1, t), and (d)E(t)obtained by solving the direct problem withN = N0 ∈ 20 (− ·−), 40 (· · ·), 80 (−−−), for Example 3.

in Figure 5.9 and Table 5.4. From Figure 5.9 it can be seen thatstable and accurate

numerical solutions are obtained. Table 5.4 also shows thatthe values of the regular-

isation parameterλdis andλGCV given by the discrepancy principle and the GCV are

similar and not very different from the optimal valueλopt given by the minimum of the

RMSE forr(t). It can also be seen that the GCV produces slightly better predictions

than the discrepancy principle.

5.5 Conclusions

The inverse problem of finding the time-dependent heat source together with the tem-

perature in the heat equation, under a non-classical dynamic boundary condition and

Chapter 5. 118

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

t

ux(1

,t)

(d)

Figure 5.8: The analytical solution (5.4.3) and the direct problem numerical solutionfrom Figures 5.7(a)–5.7(c) (—–), and numerical results of (a) r(t), (b) u(1, t), (c)ux(0, t), and (d)ux(1, t), with no regularisation, for exact data( ) and noisy datap = 1% (− · −), for Example 3.

an integral over-determination condition has been investigated. A numerical method

based on the BEM combined with the second-order Tikhonov regularisation has been

proposed together with the use of either the GCV criterion orthe discrepancy principle

for the selection of the regularisation parameter. The retrieved numerical results were

found to be accurate and stable on both smooth and non-smoothcontinuous examples.

As for the experimental validation of the proposed inverse mathematical model in

terms of bias and inverting real noisy data we defer this challenging task to possible

future work. We only remark that unlike certain applications, e.g. some significant

mismatch has been reported in [2, 35, 52] between experimental data of electromag-

netic waves propagating in a non-attenuating medium and data produced by idealised

computational simulations, in inverse heat conduction themathematical models have

Chapter 5. 119

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

t

u(1

,t)

(b)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

t

ux(1

,t)

(d)

Figure 5.9: The analytical solution (5.4.3) and the direct problem numerical solutionsfrom Figures 5.7(a)–5.7(c) (—–), and the numerical resultsof (a) r(t), (b) u(1, t), (c)ux(0, t), and (d)ux(1, t) obtained using the second-order Tikhonov regularisation withthe regularisation parameters suggested by the discrepancy method, forp ∈ 1(− ·−), 3(· · · ), 5(−−−)% noise, for Example 3.

been shown to perform much better in industrial applications with actual real measured

data, [13].

So far in previous chapters, the identification of a single time-dependent heat source

function has been the main aim. In the next two chapters, we will consider the more

challenging cases of simultaneous determination of the components of an additive or

multiplicative space- and time-dependent heat source function.

Chapter 5. 120

Table 5.4: The regularisation parametersλ given by the discrepancy principle andby the GCV, and the RMSE forr(t), u(1, t), ux(0, t) and ux(1, t), obtained usingthe BEM withN = 40 andN0 = 30 combined with the second-order Tikhonovregularisation forp ∈ 0, 1, 3, 5% noise. The optimal regularisation parameter givenby the minimum of RMSE ofr(t) is also included, for Example 3.

pparameter RMSE


1% 0 5.13E-1 3.26E-3 1.20E-2 4.19E-21% λdis=5.7E-8 1.11E-2 9.25E-4 2.89E-3 5.82E-41% λGCV =5.7E-9 1.19E-2 8.26E-4 2.76E-3 5.96E-41% λopt=2.2E-8 9.85E-3 8.06E-4 2.46E-3 5.65E-43% 0 1.53 9.86E-3 3.71E-2 1.23E-13% λdis=2.0E-6 3.13E-2 3.67E-3 9.99E-3 2.85E-33% λGCV =3.3E-7 3.01E-2 2.53E-3 8.08E-3 1.81E-33% λopt=9.0E-7 2.75E-2 2.77E-3 8.05E-3 2.07E-35% 0 1.30 1.31E-2 6.26E-2 1.03E-15% λdis=7.0E-6 7.86E-2 8.95E-3 2.68E-2 5.25E-35% λGCV =7.9E-7 4.57E-2 6.80E-3 1.72E-2 4.43E-35% λopt=9.1E-8 4.15E-2 6.68E-3 1.59E-2 4.61E-3

Chapter 6

Determination of an Additive Space-

and Time-dependent Heat Source

6.1 Introduction

Inverse source problems for the heat equation have recentlyattracted considerable in-

terest, see [1, 18, 24, 61, 64, 67] to name just a few. These studies have sought a

coefficient source function depending on either space- or time-dependent variables us-

ing various techniques. In Chapters 2–5 we have focused on the inverse problem of

finding the time-dependent coefficient source function and the temperature for the heat

equation. In this chapter we extend our study to determine inverse heat source func-

tions depending on both space and time, but which are additively separated into two

unknown coefficient source functions, namely, one component dependent on space and

another component dependent on time. The measurement/overspecified conditions are

one temperature measurement, as a function of time, at one specified interior location

and a time-average temperature throughout the space solution domain.

Since the governing partial differential equation is the linear heat equation with

constant coefficients, the preferred method of discretisation is the BEM. Even though

the inverse heat source problem is uniquely solvable, it is still ill-posed since small

121

Chapter 6. 122

errors which inherently occur in any practical measurementcause largely oscillating

solutions. To overcome this instability, in this chapter regularisation such as the TSVD

or the Tikhonov regularisation method are employed. Moreover, theL-curve method,

the GCV criterion, or the discrepancy principle are employed for the selection of the

regularisation parameter. Additionally in the case of two different regularisation pa-

rameters are considered, theL-surface criterion, [3], is utilised.

This chapter is organised as follows. In Section 6.2 the mathematical inverse for-

mulation is described and the numerical discretisation of the problem using the BEM

is presented in Section 6.3. The TSVD and the Tikhonov regularisation are described

in Section 6.4, as procedures for overcoming the instability of the solution. Finally,

Sections 6.5 and 6.6 discuss the numerical results and highlight the conclusions of this

research.


Consider the inverse problem of finding the time-dependent heat sourcer(t), the space-

dependent heat sources(x) and the temperatureu(x, t) satisfying the heat conduction

equation

ut = uxx + r(t)f(x, t) + s(x)g(x, t) + h(x, t), (x, t) ∈ DT , (6.1)


u(x, 0) = u0(x), x ∈ [0, L], (6.2)

the Dirichlet boundary conditions

u(0, t) = µ0(t), u(L, t) = µL(t), t ∈ [0, T ], (6.3)

Chapter 6. 123

and the over-determination conditions

u (X0, t) = χ(t), t ∈ [0, T ], (6.4)∫ T

0

u(x, t) dt = ψ(x), x ∈ [0, L], (6.5)

s(X0) = S0, (6.6)

whenf , g, h, u0, µ0, µL, χ, ψ are given functions,X0 is a given sensor location within

the interval(0, L), andS0 is a given value of the source functions at the given point

X0.

One can remark that the time-average temperature measurement (6.5) represents a

non-local condition/measurement. It is convenient to use in practical situations where

a local measurement of the temperature at a fixed timeT1 ∈ (0, T ], namely,

u(x, T1) =: ψT1(x), x ∈ [0, L]

contains a large amount of noise. This may be due to harsh external conditions, or

to the fact that many space measurements can, in fact, never be recorded at a fixed

instant instantaneously. If this is the case, one can have a selection of such large noise

local temperature measurements, but which on average produce a less noisy nonlocal

measurement (6.5).

The individual separate cases concerning the identification of a single time-dependent

heat sourcer(t), whens(x) is known, or the identification of a single space-dependent

heat sources(x), whenr(t) is known, have been theoretically investigated in Prilepko

and Solov’ev [44] and Prilepko and Tkachenko [45], respectively.

The unique solvability of this inverse source problem was already established by

Ivanchov [26] and it is the objective of this study to obtain astable numerical solution

of this still ill-posed problem. Note that condition (6.6) was omitted in [26], but it

should be included in order to avoid non-uniqueness and instability cases which have

been given as counterexamples in [20]. For the inverse problem (6.1)–(6.6) we have

Chapter 6. 124

the following local unique solvability theorem.

Theorem 6.2.1 Assume that the following conditions are satisfied:

(i) u0(x), ψ(x) ∈ H2+γ[0, L], µ0(t), µL(t), χ(t) ∈ H1+γ/2[0, T ], h(x, t) ∈ Hγ,γ/2(DT )

with γ ∈ (0, 1),

f independent oft andf(x) ∈ Hγ[0, L],

g independent ofx andg(t) ∈ Hγ/2[0, T ];

(ii) f(X0) 6= 0,∫ T

0

g(t) dt 6= 0,g(t)

∫ T

0g(τ) dτ

≥ 0, ∀t ∈ [0, T ];

(iii) u0(0) = µ0(0), u0(L) = µL(0), u0(X0) = χ(0),∫ T

0

χ(t) dt = ψ(X0), ψ(0) =∫ T

0

µ0(t) dt, ψ(L) =∫ T

0

µL(t) dt.

Then for sufficiently smallT > 0 there exists a unique solution(r(t), s(x), u(x, t)) ∈Hγ/2[0, T ]×Hγ[0, L]×H2+γ,1+γ/2(DT ) of the inverse problem(6.1)–(6.6).

In this theorem the functions are required to lie in Holder spaces, defined as fol-

lows:

• H i+γ1,j+γ2(DT ), with i, j = 0, 1, 2 and0 < γ1, γ2 ≤ 1, denotes the space of

continuous functions withith partial derivative with respect tox andjth partial

derivative with respect tot such that there existm1 > 0 andm2 > 0 satisfying

|∂ixu(x1, t)−∂ixu(x2, t)| ≤ m1|x1−x2|γ1 , |∂jtu(x, t1)−∂jt u(x, t2)| ≤ m2|t1−t2|γ2

for all x1, x2 ∈ [0, L] andt1, t2 ∈ [0, T ].

• Hγ(Ω), with Ω = (0, L) or (0, T ), denotes the space of continuous functions

s : Ω → R with exponent0 < γ ≤ 1 such that there existsm > 0 satisfying

|s(x1)− s(x2)| ≤ m|x1 − x2|γ for all x1, x2 ∈ Ω.

In the next Sections 6.3 and 6.4, we will demonstrate how to solve the inverse heat

source problem (6.1)–(6.6) by using a regularised BEM.

Chapter 6. 125


In the numerical process, we utilise the BEM as introduced inSection 1.3 to the heat

conduction equation (6.1). We then obtain the following boundary integral equation

η(x)u(x, t)

=

∫ t

0

[

G(x, t, ξ, τ)∂u

∂n(ξ)(ξ, τ)− u(ξ, τ)

∂G

∂n(ξ)(x, t, ξ, τ)

]

ξ∈0,L

dτ

+

∫ L

0

G(x, t, y, 0)u(y, 0) dy+

∫ L

0

∫ T

0

G(x, t, y, τ)r(τ)f(y, τ) dτdy

+

∫ L

0

∫ T

0

G(x, t, y, τ)s(y)g(y, τ) dτdy+

∫ L

0

∫ T

0

G(x, t, y, τ)h(y, τ) dτdy,

(x, t) ∈ [0, L]× (0, T ]. (6.7)

Using the same discretisation as described in the previous chapters, we obtain

η(x)u(x, t) =N∑

j=1

[A0j(x, t)q0j + ALj(x, t)qLj − B0j(x, t)h0j − BLj(x, t)hLj ]

+

N0∑

k=1

Ck(x, t)u0,k + d1(x, t) + d2(x, t) + d0(x, t). (6.8)

where

d1(x, t) =

∫ L

0

∫ T

0

G(x, t, y, τ)r(τ)f(y, τ) dτdy, (6.9)

d2(x, t) =

∫ L

0

∫ T

0

G(x, t, y, τ)s(y)g(y, τ) dτdy, (6.10)

and can be calculated by applying the piecewise constant approximations to the func-

tionsf(x, t) andr(t) as the same in (2.21), and the functionsg(x, t) ands(x) as as

g(x, t) = g(xk, t), s(x) = s(xk) =: sk, (6.11)

Chapter 6. 126

for x ∈ (xk−1, xk], k = 1, N0. Then the double integrals (6.9) and (6.10) can be

approximated as

d1(x, t) =

∫ T

0

r(τ)

∫ L

0

G(x, t, y, τ)f(y, τ) dydτ =N∑

j=1

D1,j(x, t)rj ,

d2(x, t) =

∫ L

0

s(y)

∫ T

0

G(x, t, y, τ)g(y, τ) dτdy =

N0∑

k=1

D2,k(x, t)sk,

where

D1,j(x, t) =

∫ L

0


D2,k(x, t) =1

2

∫ T

0

g(xk, t)H(t− τ)

[

erf

(

x− xk−1

2√t− τ

)

− erf

(

x− xk

2√t− τ

)]

dτ,

These integrals are evaluated using Simpson’s rule for numerical integration. With

these approximations, the integral equation (6.8) becomes

η(x)u(x, t) =

N∑

j=1


+

N0∑

k=1

Ck(x, t)u0,k +N∑

j=1

D1,j(x, t)rj +

N0∑

k=1

D2,k(x, t)sk

+

N∑

j=1

D0,j(x, t). (6.12)

Applying the equation (6.12) at the boundary nodes(0, ti) and (L, ti) for i = 1, N

yields the system of2N linear equations

Aq¯−Bh

¯+ Cu

¯0+D1r

¯+D2s

¯+ d

¯= 0

¯, (6.13)

Chapter 6. 127

where matricesA, B, C and vectors q¯, h

¯, u

¯0, d

¯are defined the same as in (2.25), and

D1 =

D1,j(0, ti)

D1,j(L, ti)

2N×N

, D2 =

D2,k(0, ti)

D2,k(L, ti)

2N×N0

, s¯=[

sk

]

N0

.

Here, the boundary temperature h¯

is known by the boundary condition (6.3), i.e.

h¯=

u(0, tj)

u(L, tj)

2N

=

µ¯0µ¯L

2N

=: µ¯. (6.14)

Therefore from (6.13), we obtain

q¯= A−1

(

Bµ¯− Cu

¯0−D1r

¯−D2s

¯− d

¯

)

. (6.15)

To determine r¯

and s¯, the conditions (6.4)–(6.6) are imposed. Firstly, we consider

the interior points(X0, ti) for i = 1, N which can be written as

χi := χ(ti) = u(X0, ti), i = 1, N. (6.16)

Applying the interior points above to the equation (6.12) can give rise to the following

linear system ofN equations:

AIq¯−BIµ

¯+ CIu

¯0+DI

1r¯+DI

2s¯+ d

¯I = χ

¯, (6.17)

where

AI =[

A0j(X0, ti) ALj(X0, ti)]

N×2N, BI =

[

B0j(X0, ti) BLj(X0, ti)]

N×2N,

CI =[

Ck(X0, ti)]

N×N0

, DI1 =

[

D1,j(X0, ti)]

N×N, DI

2 =[

D2,k(X0, ti)]

N×N0

,

d¯I =

[

∑Nj=1D0,j(X0, ti)

]

N, χ

¯=[

χi

]

N.

Whereas the time-integral condition (6.5) is approximatedby the midpoint numerical

Chapter 6. 128

integration, as in the parallel way as (3.11), atx = xk for k = 1, N0, then we obtain

T

N

N∑

i=1

u(xk, ti) =

∫ T

0

u(xk, t) dt = ψ(xk) =: ψk for k = 1, N0. (6.18)

Using (6.12), equation (6.18) yields

T

N

N∑

i=1

[

AIIi q¯− BII

i µ¯+ CII

i u¯0

+DII1,ir¯

+DII2,is¯

+ d¯IIi

]

= ψ¯, (6.19)

where

AIIi =[

A0j(xk, ti) ALj(xk, ti)]

N0×2N, BII

i =[

B0j(xk, ti) BLj(xk, ti)]

N0×2N,

CIIi =

[

Ck(xk, ti)]

N0×N0

, DII1,i =

[

D1,j(xk, ti)]

N0×N, DII

2,i =[

D2,k(xk, ti)]

N0×N0

,

d¯IIi =

[

∑Nj=1D0,j(xk, ti)

]

N0

, ψ¯=[

ψk

]

N0

.

Finally, we consider the condition (6.6). Since we have usedthe space midpoint

discretisation, we then approximateS0 at the given pointX0 ∈ (0, L) as

S0 = s(X0) ≈s(xk∗) + s(xk∗+1)

2, (6.20)

where indexk∗ ∈ 1, . . . , N0 − 1 satisfiesxk∗ ≤ X0 < xk∗+1.

Now the approximate solutions r¯

and s¯

can be found by eliminating q¯

from (6.13)

and combining expressions (6.17), (6.19), and (6.20), to obtain, after some manipula-

tions, a linear system of(N +N0+1) equations with(N +N0) unknowns as follows:

Xw¯= y

¯, (6.21)

Chapter 6. 129

where

X =

AIA−1D1 −DI1 AIA−1D2 −DI

2

T

N

N∑

i=1

(AIIi A−1D1 −DII

1,i)T

N

N∑

i=1

(AIIi A−1D2 −DII

2,i)

0 . . . 0 0 . . . 0 12

120 . . . 0

(N+N0+1)×(N+N0)

,

y¯=

−χ+ AIA−1(Bµ¯− Cu

¯0− d

¯)− BIµ

¯+ CIu

¯0+ d

¯I

−ψ +T

N

N∑

i=1

(

AIIi A−1(Bµ

¯− Cu

¯0− d

¯)− BII

i µ¯+ CII

i u¯0

+ d¯IIi

)

S0

N+N0+1

,

and w¯=

r¯s¯

N+N0

.

Since the problem is ill-posed, then the system of equations(6.21) is ill-conditioned.

In the next section, we will deal with this ill-conditioningusing regularisation in order

to obtain a stable solution.

6.4 Regularisation

In practice, the measured data is unavoidably contaminatedby unplanned error. In

order to model this, we add noise into the input functionsχ(t) andψ(x) representing

the over-determination conditions (6.4) and (6.5) as follows:

χ¯ǫ = χ

¯+ random(′Normal′, 0, σχ, 1, N), (6.22)

and

ψ¯ǫ = ψ

¯+ random(′Normal′, 0, σψ, 1, N0), (6.23)

with the standard deviationsσχ andσψ to be taken as

σχ = p× maxt∈[0,T ]

|χ(t)|, and σψ = p× maxx∈[0,L]

|ψ(x)|, (6.24)

Chapter 6. 130

respectively. Note that the measurement (6.6) is already contaminated by error due to

the approximation made in (6.20).

If we consider the contamination of the right-hand side of equation (6.21) as‖y¯ǫ−

y¯‖ ≈ ǫ, then the direct least-squares solution w

¯= (XTX)−1XTy

¯ǫ will be unstable. To

overcome this instability, regularisation method needs tobe utilised. In this study, we

employ either the TSVD or the Tikhonov regularisation methods.

We first consider the use of the TSVD method as a regularisation procedure. To

use this method we use the[U,Σ,V]=svds(X,Nt) command in MATLAB, as we have

used previously in Chapter 3. In order to indicate the appropriate truncation levelNt,

theL-curve criterion, the GCV method, and the discrepancy principle are utilised.

Alternatively, the Tikhonov regularisation is another wayof obtaining a stable so-

lution of the ill-conditioned system of equations (6.21) which based on minimising the

regularised linear least-squares objective function

‖Xw¯− y

¯ǫ‖2 + λr‖R(1)r‖2 + λs‖R(2)s‖2 (6.25)

whereR(1), R(2) are (differential) regularisation matrices corresponding to a regular-

isation parameterλr, λs > 0, respectively. Solving (6.25) one obtains the regularised

solution

w¯λr ,λs

=(

XTX +RTR)−1

XTy¯ǫ. (6.26)

where the matrixR represents a block matrix of upper-left subblockλrR(1) and lower-

right subblockλsR(2). Initially, we takeλ := λr = λs and consider theL-curve

criterion, the GCV method, and the discrepancy principle aschoices for indicating

the single regularisation parameterλ. Note that both theL-curve and the GCV are

heuristic methods because they do not require the knowledgeof the level of noiseǫ.

More rigorously, one can use the discrepancy principle, [40], which selectsλ such that

‖Xw¯λ

− y¯ǫ‖ ≈ ǫ. (6.27)

Chapter 6. 131

If we allow for general multiple regularisation parametersλr andλs in (6.25) then, for

their selection one could employ theL-surface criterion, [3], which plots the residual

‖Xw¯λr ,λs

− y¯ǫ‖ versus‖R(1)r‖ and‖R(2)s‖ for various values ofλr andλs.


In this section, we present two benchmark test examples in order to test the accuracy

of the approximate solutions. We are using the RMSE forr(t) as defined in (2.49)

whereas the RMSE fors(x) can be defined as

RMSE(s(x)) =

√

√

√

√

L

N0

N0∑

k=1

(sexact(xk)− snumerical(xk))2. (6.28)

6.5.1 Example 1

In the first example, we consider a smooth benchmark test withT = L = 1, X0 = 12

and the input data

u0(x) = u(x, 0) = x2, µ0(t) = u(0, t) = 0, µL(t) = u(1, t) = et,

χ(t) = u(12, t) =

et

4, ψ(x) =

∫ 1

0

u(x, t) dt = x2(e− 1),

S0 = s(12) = 1, f(x, t) = ex, g(x, t) = t+ 1,

h(x, t) = (x2 − 2)et − t2ex − (t + 1) sin(πx).

(6.29)

One can check that the conditions of Theorem 6.2.1 are satisfied hence the inverse

source problem (6.1)–(6.6) with the data (6.29) has a uniquesolution. It can easily be

verified through direct substitution that this solution is given by

u(x, t) = x2et, r(t) = t2, s(x) = sin(πx). (6.30)

As mentioned in Section 6.2, the inverse heat source problem(6.1)–(6.6) is ill-

Chapter 6. 132

posed since small errors in the measured data (6.4)–(6.6) cause large errors in the

solution. In order to quantify the degree of ill-conditioning we calculate the condition

number of the matrixX. The condition numbers forN = N0 ∈ 20, 40, 80 and

X0 ∈ 14, 12, 34 are shown in Table 6.1. In addition, the normalised singularvalues of

the matrixX are displayed in Figure 6.1, and the rapidly decreasing values indicate that

the system of equations (6.21) is ill-conditioned. Lookingat the columns of Table 6.1

it can be seen that the condition number only slightly decreases asX0 increase, hence

we do not expect the numerical results to be significantly influenced by the choice of

X0 within some interval[14, 34] away from the end pointsx = 0 andx = L = 1. Of

course, asX0 gets closer to the boundary pointx = 0 orx = L then the specification of

the interval temperature measurement (6.4) resembles a heat flux prescription, namely

ux(0, t) = limX0ց0

u(X0, t)− u(0, t)

X0, or ux(L, t) = lim

X0րL

u(X0, t)− u(L, t)

X0 − L.

However, this newly generated inverse problem in which Cauchy data are specified at

x = 0 or x = L is not addressed herein and it is deferred to a future work.

0 20 40 60 80 100 120 140 16010

−5

10−4

10−3

10−2

10−1

100

Nor

mal

ised

Sin

gula

r V

alue

s

2N

Figure 6.1: The normalised singular values of matrixX for N = N0 ∈ 20, 40, 80andX0 ∈ 1

4(− · −), 1

2(· · · ), 3

4(−−−), for Example 1.

In what follows, the numerical results are illustrated for afixed discretisationN =

N0 = 40 andX0 =12.

Chapter 6. 133

Table 6.1: The condition numbers of the matrixX in equation (6.21) for variousN =N0 ∈ 20, 40, 80 andX0 ∈ 1

4, 12, 34, for Example 1.

N = N0 20 40 80X0 = 1/4 1.94E+3 9.07E+3 5.04E+4X0 = 1/2 1.97E+3 7.51E+3 4.06E+4X0 = 3/4 1.93E+3 6.50E+3 3.33E+4

Exact Data

We consider first the case of exact data, i.e.p = 0 in (6.24). We directly solve

the linear system of equations (6.21) with the untruncated SVD method, and display

the numerical solutions forr(t), s(x), ux(0, t), andux(1, t) in Figures 6.2(a)–6.2(d),

respectively. From these figures, it can be seen that the solutions for r(t) ands(x)

are inaccurate, but the fluxesux(0, t) andux(1, t) are stable and accurate with small

RMSEs of 9.32E-3 and 4.58E-2, respectively, see Table 6.2. This is somewhat to be

expected since the inverse problem is ill-posed. Hence, regularisation is required to

overcome this instability.

For this, we utilise the TSVD and the Tikhonov regularisation of orders zero, one,

and two. The selection method of the regularisation parameters is first considered.

TheL-curves of the TSVD and the Tikhonov regularisations are presented in Figures

6.3(a) and 6.4(a), respectively. It can be seen that there isnoL-shape obtained for ei-

ther the TSVD, ZOTR, or FOTR, whereas the SOTR shows more clearly anL-corner

at λL=1E-1. Alternatively, the GCV method is utilised as anotherchoice for the reg-

ularisation parameter, as shown in Figure 6.3(b). The minimum of the GCV function

suggestsNt = 56 to be the truncation number for the TSVD, whilst for the Tikhonov

regularisation which is displayed in Figure 6.4(b), the minima indicate the parameters

λGCV =1.0E-7, 1.2E-7, and 4.5E-8 for ZOTR, FOTR and SOTR, respectively. Note

that for the exact data,ǫ ≈ 0 and the discrepancy principle cannot be employed. With

the GCV selection for the regularisation parameters determined from Figures 6.3(b)

and 6.4(b), the TSVD and the Tikhonov regularisation results are shown in Figure 6.5.

Compared to Figure 6.2, one can see that the instability of the numerical solutions is

Chapter 6. 134

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

s(x

)(b)

0 0.2 0.4 0.6 0.8 1−0.02

−0.015

−0.01

−0.005

0

0.005

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 12

2.5

3

3.5

4

4.5

5

5.5

t

ux(1

,t)

(d)

Figure 6.2: The analytical (—–) and numerical results(− · −) of (a) r(t), (b) s(x), (c)ux(0, t), and (d)ux(1, t) obtained using the SVD for exact data, for Example 1.

not alleviated. We then employ another choice of the regularisation parameter based on

theL-curve method. This suggestsλL=1E-1 for the SOTR displayed in Figure 6.4(b).

Then with this choice forλ we obtain the stable and accurate numerical results shown

in Figure 6.6 and Table 6.2.

Noisy Data

Next, the case of noise contamination with percentagep = 1% is considered by adding

random noise into the input functionsχ(t) andψ(x) in (6.29), as in (6.22) and (6.23),

respectively. It is of crucial importance to utilise the regularisation in this case, and se-

lecting the regularisation parameter is the first step of theregularisation process. Here,

Chapter 6. 135

10−4

10−3

10−2

10−1

100

4.5

4.7

4.9

5.1

5.3

Nt=2

Nt=3

Nt=4

Nt=5

Nt=6

Nt=7

Nt=8,..,37

Nt=38,..,46

Nt=47,..,80

‖Xw¯Nt

− y¯

ǫ‖

‖R

w ¯N

t‖

(a)

10 20 30 40 50 60 70 80

10−10

10−9

10−8

10−7

Nt = 56

Nt

GC

V(N

t)

Nt=20

Nt=80

(b)

Figure 6.3: (a) TheL-curve and (b) the GCV function obtained by the TSVD for exactdata, for Example 1.

10−4

10−3

10−2

10−1

100

101

10−5

10−4

10−3

10−2

10−1

100

101

λ = 1E−8

‖Xw¯λ

− y¯

ǫ‖

‖R

w ¯λ‖

λ = 1E−8

λ = 1E−1

(a)

10−9

10−8

10−7

10−6

10−5

10−11

10−10

10−9

λ = 4.5E−8

λ

GC

V(λ

)

λ = 1.2E−7

λ = 1.0E−7

(b)

Figure 6.4: (a) TheL-curve, and (b) the GCV function, obtained by the ZOTR(−·−),FOTR(· · · ), and SOTR(−−−) for exact data, withλ = λr = λs, for Example 1.

theL-curve method and the discrepancy principle are employed ascriteria for choos-

ing the regularisation parameters. These are displayed in Figures 6.7 and 6.8 using the

TSVD and the Tikhonov regularisation, respectively. The suggested parameters are

given in Table 6.2. Figure 6.9 presents all results obtainedusing the TSVD and the

Tikhonov regularisation of orders zero, one, and two with the regularisation parame-

ters suggested by the discrepancy principle, see Table 6.2.Looking more closely at

Figure 6.9(a), it can be seen that the approximate solutionsfor r(t) obtained by the

Chapter 6. 136

first- and the second-order Tikhonov regularisation are reasonably stable, whereas the

numerical solution fors(x), as shown in Figure 6.9(b) is rather in accurate.

We consider the second-order Tikhonov regularisation withthe regularisation pa-

rameter suggested by theL-curve methodλL=10 and obtain the results shown in Fig-

ure 6.11. After analyzing this numerical solution, it can beclearly observed that we

cannot obtain accurate solutions for bothr ands usingλr = λs. Therefore, the case

λr 6= λs is considered and theL-surfaces are shown in Figure 6.10. On the plane of

logarithm of residual norm,log ‖Xw¯λ

− y¯ǫ‖, versus logarithm of the second deriva-

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

s(x

)

(b)

0 0.2 0.4 0.6 0.8 1−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 12

2.5

3

3.5

4

4.5

5

5.5

t

ux(1

,t)

(d)

Figure 6.5: The analytical (—–) and numerical results of (a)r(t), (b)s(x), (c)ux(0, t),and (d)ux(1, t) obtained using the TSVD(− + −), ZOTR (− · −), FOTR (· · · ),and SOTR(−−−) with regularisation parameters suggested by the GCV function ofFigure 6.3(b) and 6.4(b) for exact data, for Example 1.

Chapter 6. 137

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

s(x

)

(b)

0 0.2 0.4 0.6 0.8 1−0.005

0

0.005

0.01

0.015

0.02

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 12

2.5

3

3.5

4

4.5

5

5.5

t

ux(1

,t)

(d)

Figure 6.6: The analytical (—–) and numerical results(− · −) of (a) r(t), (b) s(x), (c)ux(0, t), and (d)ux(1, t) obtained using the SOTR with the regularisation parameterλL=1E-1 suggested by theL-curve of Figure 6.4(a) for exact data, for Example 1.

tive of r, log ‖R(1)rλr‖, forms anL-shaped corner atλr=1E+1, whileλs=1 is based

around the area of theL-corner on the plane oflog ‖Xw¯λ

− y¯ǫ‖ versuslog ‖R(2)rλs‖.

However, the numerical solution fors(x) obtained using the parametersλr,L = 10,

λs,L = 1 suggested by theL-surface method, is still inaccurate. We finally use the trial

and error process to seek out the appropriated regularisation parameters, and found that

regularisation parametersλr,opt=8 andλs,opt=5.2E-2 can yield an accurate and stable

numerical solution, see Figure 6.11. Nevertheless, more research has to be undertaken

in the future for the selection of appropriate multiple regularisation parameters, [12].

for completeness, the RMSE of all results which we have mentioned so far are detailed

Chapter 6. 138

10−2

10−1

100

100

101

102

103

N t=2N t

=3N t=5

N t=4

Nt=53,54

Nt=55,..,57

Nt=66,67

N t=41,..,43N t=44,..,48N t

=49

Nt=62,..,65

Nt=50,51

Nt=58,..,61

Nt=68,..,76

Nt=77,78N

t=79N

t=80

‖Xw¯Nt

− y¯

ǫ‖

‖R

w ¯N

t‖

Nt=6,..,40

(a)

0 10 20 30 40 50 60 70 8010

−2

10−1

100

101

ε = 0.11

Nt = 14

Nt

‖X

w ¯N

t−

y ¯ǫ‖

(b)

Figure 6.7: (a) TheL-curve and (b) the discrepancy principle obtained using theTSVDfor noisy inputp = 1%, for Example 1.

0.06 0.08 0.1 0.2 0.410

−6

10−4

10−2

100

102

104

λ = 1E+1λ = 1E+2

‖Xw¯λ

− y¯

ǫ‖

‖R

w ¯λ‖

λ = 1E−1λ = 1

λ = 1E−4λ = 1E−3

(a)

10−5

10−4

10−3

10−2

10−1

100

101

102

10−1

100

λ = 1.5

λ

‖X

w ¯λ−

y ¯ǫ‖

ε = 0.11

λ = 2.8E−2λ = 1.3E−3

(b)

Figure 6.8: (a) TheL-curve and (b) the discrepancy principle obtained using theZOTR(− · −), FOTR(· · · ), and SOTR(−−−) for noisy inputp = 1%, with λ = λr = λs,for Example 1.

in Table 6.2.

6.5.2 Example 2

In example 1, the case of smooth source functions has been investigated and it can

retrieved the instability with the use of BEM together with the regularisation based on

either the TSVD and the Tikhonov regularisation. In this example, we are considering

Chapter 6. 139

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

s(x

)

(b)

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11.5

2

2.5

3

3.5

4

4.5

5

5.5

6

t

ux(1

,t)

(d)

Figure 6.9: The analytical (—–) and numerical results of (a)r(t), (b)s(x), (c)ux(0, t),and (d)ux(1, t) obtained using the TSVD(− + −) with Nt = 14, and the ZOTR(− · −), FOTR(· · · ), SOTR(−−−) with regularisation parameters suggested by thediscrepancy principle of Figure 6.8(b) for noisy inputp = 1%, for Example 1.

Chapter 6. 140

−2.35 −2.3 −2.25 −2.2 −2.15 −2.1 −2.05 −2−20

−10

0

10

−20

−15

−10

−5

0

5

log(‖Xw¯λ

− y¯

ǫ‖)

log(‖R(2)s¯λ‖)

log(‖R

(1)r ¯λ‖)

(a)

−2.35 −2.3 −2.25 −2.2 −2.15 −2.1 −2.05 −2 −1.95−20

−15

−10

−5

0

5

log(‖Xw¯λ

− y¯

ǫ‖)

log(‖R

(1)r ¯λ‖)

λr=1E−5

λr=1E+1

(b)

−2.35 −2.3 −2.25 −2.2 −2.15 −2.1 −2.05 −2 −1.95−20

−15

−10

−5

0

5

log(‖Xw¯λ

− y¯

ǫ‖)

log(‖R

(2)s ¯λ‖)

λs=1E−5

λs=1

(c)

Figure 6.10: TheL-surface on (a) a three-dimensional plot, (b) plane oflog ‖Xw¯λ−y

¯ǫ‖

versuslog ‖R(1)r¯λ‖, and (c) plane oflog ‖Xw

¯λ− y


¯λ‖, obtained

using the SOTR for noisy inputp = 1%, for Example 1.

Chapter 6. 141

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

s(x

)

(b)

0 0.2 0.4 0.6 0.8 1−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 11.5

2

2.5

3

3.5

4

4.5

5

5.5

t

ux(1

,t)

(d)

Figure 6.11: The analytical (—–) and numerical results of (a) r(t), (b) s(x), (c)ux(0, t), and (d)ux(1, t) obtained using the SOTR with regularisation parameters sug-gested by theL-curve criterionλL = λr = λs = 10 (− − −), theL-surface method(λr,L, λs,L)=(10,1)(−+−), and the trial and error(λr,opt, λs,opt)=(8,5.2E-2)(− ∗ −),for noisy inputp = 1%, for Example 1.

Chapter 6. 142

Table 6.2: The RMSE forr(t), s(x), ux(0, t), andux(1, t) obtained using the SVD,TSVD, ZOTR, FOTR, and SOTR, forp ∈ 0, 1%, for Example 1.

Method p ParameterRMSE

r(t) s(x) ux(0, t) ux(1, t)SVD 0 - 1.47E-1 2.55E-1 9.32E-3 4.58E-2

TSVD 0 Nt=56 1.17E-1 2.03E-1 2.94E-3 3.57E-2ZOTR 0 λGCV =1.0E-7 1.20E-1 2.02E-1 3.53E-3 3.65E-2FOTR 0 λGCV =1.2E-7 7.62E-2 1.70E-1 3.68E-3 4.35E-2SOTR 0 λGCV =4.5E-8 7.96E-2 1.85E-1 6.48E-3 4.70E-2SOTR 0 λL=1.0E-1 8.70E-3 2.81E-2 1.27E-2 3.09E-3

SVD 1% - 1.62E+1 1.01E+2 2.84 2.48E-1TSVD 1% Nt=14 2.04E-1 1.77E-1 2.29E-2 5.83E-2ZOTR 1% λdis=1.3E-3 1.87E-1 1.83E-1 2.32E-2 4.87E-2FOTR 1% λdis=2.8E-2 1.28E-1 3.50E-1 1.05E-1 7.91E-2SOTR 1% λdis=1.5 9.72E-2 2.65E-1 8.05E-2 5.58E-2SOTR 1% λL=10 1.61E-1 4.23E-1 1.20E-1 9.36E-2SOTR 1% λr=10,λs=1 7.93E-2 2.18E-1 6.81E-2 4.40E-2SOTR 1% λr=8,λs=5.2E-2 1.92E-3 5.34E-2 1.00E-2 6.39E-3

in more severe case with the non-smooth source functions. Let T = L = 1, X0 = 12

and the input data

u0(x) = µ0(t) = µL(t) = 0, S0 = s(12) =

1

4,

χ(t) = u(12, t) = t2 sin(1

4), ψ(x) =

∫ 1

0

u(x, t) dt =sin(x− x2)

3,

f(x, t) = x, g(x, t) = et,

h(x, t) = (2t+ t2(1− 2x)2) sin(x− x2) + 2t2 cos(x− x2)

−x|t− 12| − et|x− 3

4|.

(6.31)

Note that the input data (6.31) satisfy the conditions of Theorem 6.2.1 to ensure the

existence and uniqueness of solution of the inverse problem(6.1)–(6.5). In fact, the

exact solution is given by

u(x, t) = t2 sin(x− x2), r(t) = |t− 12|, s(x) = |x− 3

4|.

Chapter 6. 143

0 20 40 60 80 100 120 140 16010

−5

10−4

10−3

10−2

10−1

100

Cond = 1.54E+4N

orm

alis

ed S

ingu

lar

Val

ues

2N

Cond = 8.69E+4

Cond = 3.46E+3

Figure 6.12: The normalised singular values of matrixX for N = N0 = 20 (− · −),N = N0 = 40 (· · · ), andN = N0 = 80 (−−−), for Example 2.

This is a more severe test example than Example 1 since the source componentsr(t)

ands(x) are not smooth functions.

We have calculated the condition numbers of the matrixX and obtained the condi-

tion numbers 3.46E+3, 1.54E+4, and 8.69E+4 forN = N0 = 20, 40, and 80, respec-

tively. Moreover, the corresponding normalised singular values are shown in Figure

6.12. In Example 2, the condition numbers of the matrixX are not much different

from the condition numbers for Example 1. Then we expect to solve this inverse prob-

lem by using wither the TSVD or the Tikhonov regularisation as means to reduce the

instability of the solution. Here, we fixN = N0 = 40 andX0 =12.

Exact Data

First we have tried the TSVD, ZOTR, FOTR and SOTR with the regularisation pa-

rameter given by the GCV function. This yieldsNt = 65, λGCV =2.9E-8, 3.2E-8, and

8.3E-9, respectively. But we have found that the solutions for r(t) ands(x) are not

so accurate. We then considered theL-curve method for choosing the regularisation

parameter. Figures 6.13(a) and 6.13(b) display theL-curves for the TSVD and the

Tikhonov regularisation, respectively. The same as theL-curve in Example 1, anL-

shape is obtained only when using the SOTR with suggests anL-corner aroundλ=1E-4

to 1E-3. In particular, forλL=1E-4 we obtain the stable solutions presented in Figure

Chapter 6. 144

10−5

10−4

10−3

10−2

10−1

100

0.4

0.6

0.8

1.0

1.5

2.0

2.5

3.0

3.5

Nt=1

Nt=2,3

Nt=4

Nt=5,.,10

N t=11,..,24N t

=25,..,34N t=35,..,41

N t=42,..,45

N t=46,..,4

9

N t=51,..,5

4

N t=56,..,5

9

N t=60,..,8

0

‖Xw¯Nt

− y¯

ǫ‖

‖R

w ¯N

t‖

(a)

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

101

λ = 1E−5

λ = 1E−4

λ = 1E−3λ = 1E−2

λ = 1E−1

λ = 1

λ = 1E+1

λ = 1E+2

‖Xw¯λ

− y¯

ǫ‖

‖R

w ¯λ‖

λ = 1E−5 λ = 1E−4 λ = 1E−3 λ = 1E−2λ = 1E−1

λ = 1

λ = 1E+1

λ = 1E+2

λ = 1E−4λ = 1E−3λ = 1E−2λ = 1E−1

λ = 1

λ = 1E+1

λ = 1E+2

(b)

Figure 6.13: TheL-curve obtained using (a) the TSVD and (b) the ZOTR(− · −),FOTR(· · · ), and SOTR(−−−) with λ = λr = λs, for exact data, for Example 2.

6.14 and Table 6.3. The untruncated SVD, i.e.Nt = 80, whose numerical results are

also included is not so accurate and stable in retrieving thefunctionsr(t) ands(x).

Table 6.3: The RMSE forr(t), s(x), ux(0, t), andux(1, t) obtained using the SVD,TSVD, ZOTR, FOTR, and SOTR, forp ∈ 0, 1%, for Example 2.

Method p ParameterRMSE

r(t) s(x) ux(0, t) ux(1, t)SVD 0 - 1.15E-1 4.12E-2 2.05E-3 6.95E-4

TSVD 0 Nt=65 1.17E-1 7.92E-2 1.15E-1 4.12E-2ZOTR 0 λGCV =2.9E-8 1.67E-1 6.63E-2 5.21E-3 2.19E-3FOTR 0 λGCV =3.2E-8 8.94E-2 2.99E-2 2.12E-3 6.01E-4SOTR 0 λGCV =8.3E-9 9.20E-2 3.13E-2 2.10E-3 7.61E-4SOTR 0 λL=1.0E-4 5.88E-3 8.94E-3 2.39E-3 1.04E-3

SVD 1% - 5.31E+1 8.91E+1 2.88 2.42E-1TSVD 1% Nt=10 2.16E-1 2.37E-1 1.15E-1 9.05E-2ZOTR 1% λdis=7.3E-4 1.20E-1 2.12E-1 1.15E-1 4.72E-2FOTR 1% λdis=3.2E-2 1.66E-1 6.77E-2 2.21E-2 2.51E-2SOTR 1% λdis=2.3 5.78E-2 7.90E-2 9.98E-3 4.95E-2SOTR 1% λL=1 9.24E-2 6.63E-2 1.55E-2 4.04E-2SOTR 1% λr,L=1,λs,L=10 3.96E-2 1.13E-1 4.67E-3 6.40E-2SOTR 1% λr,opt=2.2,λs,opt=5.9 2.37E-2 1.01E-1 3.42E-3 5.98E-2

Chapter 6. 145

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

s(x

)

(b)

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

t

ux(1

,t)

(d)

Figure 6.14: The analytical (—–) and numerical results of (a) r(t), (b) s(x), (c)ux(0, t), and (d)ux(1, t) obtained using the SVD(− · −) and the SOTR(−o−) withthe regularisation parameterλL=1E-4 suggested by theL-curve of Figure 6.13(b) forexact data, for Example 2.

Noisy Data

When noise is present in the measured dataχ(t) andψ(x), the regularisation with

an appropriate parameter has to be carefully considered. Here we have tried solving

the perturbed problem withp = 1% noisy input by using the TSVD, ZOTR, FOTR,

and SOTR, with the regularisation parameter given by the discrepancy principle. This

yieldsNt = 10, λdis=7.3E-4, 3.2E-2, and 2.3, respectively. Although the discrepancy

principle is a rigorous method which uses the knowledge of noise, the RMSE errors

displayed in Table 6.3 are quite large. Alternatively, we consider theL-curve method

Chapter 6. 146

10−2

10−1

100

10−1

100

101

102

Nt=1

Nt=2,..,4N

t=5,..,30

Nt=31,..,45

Nt=46,..,49

Nt=50,..,64

Nt=65,66

Nt=67,..,75

Nt=76,77

Nt=78,79N

t=80

‖Xw¯Nt

− y¯

ǫ‖

‖R

w ¯N

t‖

(a)

0.08 0.1 0.210

−6

10−4

10−2

100

102

104

λ = 1E−2

λ = 1

λ = 1E+1

‖Xw¯λ

− y¯

ǫ‖

‖R

w ¯λ‖

λ = 1E−3λ = 1E−4

λ = 1E−1

(b)

Figure 6.15: TheL-curve obtained using (a) the TSVD and (b) the ZOTR(− · −),FOTR (· · · ), and SOTR(− − −) with λ = λr = λs, for noisy inputp = 1%, forExample 2.

for the choice of regularisation parameter displayed in Figure 6.15. This suggests the

appropriate parameters asNt between 5 and 30,λL=1E-4, 1E-2, and 1 for the TSVD,

ZOTR, FOTR, and SOTR,, respectively. We then solved the inverse problem with these

parameters and found that the numerical results obtained using the TSVD, ZOTR and

FOTR, are not so accurate. Whereas the SOTR yields a more accurate solution, as

shown in Figure 6.17 with dashed line. Hence, as in Example 1,the case ofλr 6= λs

needs to be considered by using theL-surface method for choosing the appropriate

regularisation parameters. Figures 6.16 displays theL-surface which selectsλr,L=10

andλs,L=1, and the results obtained using the SOTR with these parameters are shown

in Figure 6.17. Furthermore, the regularisation parameters selected by the trial and

error have also been considered and these results have also been included in Figure

6.17. The accurate retrieval ofr(t) is possible, but fors(x) this is less accurate.

6.6 Conclusions

This chapter has presented a numerical approach to the simultaneous numerical de-

termination of the space- and the time-dependent coefficient source functions of an

Chapter 6. 147

−2.25−2.2

−2.15−2.1

−20−15

−10−5

0−20

−15

−10

−5

0

log(‖Xw¯λ

− y¯

ǫ‖)log(‖R(2)s

¯λ‖)

log(‖R

(1)r ¯λ‖)

(a)

−2.25 −2.2 −2.15 −2.1−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

log(‖Xw¯λ

− y¯

ǫ‖)

log(‖R

(1)r ¯λ‖)

λr=1E−3

λr=1

λr=10

(b)

−2.25 −2.2 −2.15 −2.1−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

log(‖Xw¯λ

− y¯

ǫ‖)

log(‖R

(2)s ¯λ‖)

λs=1E−3

λs=1

λs=10

(c)

Figure 6.16: TheL-surface on (a) a three-dimensional plot, (b) plane oflog ‖Xw¯λ

−y¯ǫ‖

versuslog ‖R(1)r¯λ‖, and (c) plane oflog ‖Xw

¯λ− y


¯λ‖, obtained

using the SOTR for noisy inputp = 1%, for Example 2.

inverse heat conduction problem with Dirichlet boundary conditions together with

specified interior temperature measurement and time-integral condition, as the over-

determination conditions.

The numerical discretisation was based on the BEM together with either the TSVD,

or the Tikhonov regularisation. Additionally, various methods for choosing the regu-

larisation parameters have been utilised. The numerical results presented show that

accurate and stable numerical solutions can be achieved provided that the regularisa-

tion parameters are appropriately selected. The two-parameter selection has proved

to be difficult, as some of our numerical results obtained using several criteria, e.g.

Chapter 6. 148

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t

r(t

)

(a)

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

x

s(x

)(b)

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

ux(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t

ux(1

,t)

(d)

Figure 6.17: The analytical (—–) and numerical results of (a) r(t), (b) s(x), (c)ux(0, t), and (d)ux(1, t) obtained using the SOTR with regularisation parameters sug-gested by theL-curve criterionλL = λr = λs = 1 (− − −), theL-surface method(λr,L, λs,L)=(1,10)(− + −), and the trial and error(λr,opt, λs,opt)=(2.2,5.9)(− ∗ −),for noisy inputp = 1%, for Example 2.

discrepancy principle, GCV,L-curve,L-surface, have shown. Nevertheless, more re-

search has to be undertaken in the future for the selection ofmultiple regularisation

parameters, [12].

In the next chapter we will consider reconstructing multiplicative space- and time-

dependent heat sources.

Chapter 7

Determination of Multiplicative Space-

and Time-dependent Heat Sources

7.1 Introduction

In the previous chapter, we have investigated the reconstruction of an additive source

of the formr(t)f(x, t)+ s(x)g(x, t). In this chapter, we consider the reconstruction of

a multiplicative source of the formr(t)s(x), in which bothr(t) ands(x) are unknown

functions. In contrast to the previously investigated linear reconstruction of the additive

source, Chapter 6, this new inverse source problem formulation is more difficult to

solve because it now becomes nonlinear. Moreover, its ill-posedness with respect to

small errors in the input data being blown up in the output source solution adds even

further difficulty.

The existence and uniqueness of the sourcesr(t), s(x) and the temperatureu(x, t)

of the inverse problem were already established in [47]. In this chapter, we consider

obtaining a stable solution by using the BEM together with a nonlinear minimisation.

The plan of the chapter is as follows. In Section 7.2, we give the mathematical for-

mulation of the inverse multiplicative source problem and state its unique solvability.

In Section 7.3, we describe the numerical discretisation ofthe problem based on the

149

Chapter 7. 150

BEM, whilst in Section 7.4 we introduce the inverse method for obtaining the solution

based on a nonlinear least-squares minimisation. Section 7.5 presents and discusses

numerical results and illustrates the need for employing regularisation in order to sta-

bilise the solution. Finally, Section 7.6 presents the conclusions of the study.


Consider the following inverse initial-boundary value problem of finding the temper-

atureu(x, t) and the multiplicatively separable source functionF (x, t) := r(t)s(x)

satisfying the heat equation

ut = uxx + r(t)s(x), (x, t) ∈ DT , (7.1)


u(x, 0) = u0(x), x ∈ [0, L], (7.2)

the homogeneous Neumann boundary conditions

ux(0, t) = ux(L, t) = 0, t ∈ [0, T ], (7.3)

the additional temperature measurement

u(X0, t) = χ(t), t ∈ [0, T ], (7.4)

at a fixed sensor locationX0 ∈ (0, L), and

u(x, T ) = β(x), x ∈ [0, L], (7.5)

at the ‘upper-base’ final timet = T . Conditions (7.3) express that the ends0, L of

the finite slab(0, L) are insulated. In order to avoid trivial non-uniqueness represented

Chapter 7. 151

by the identityr(t)s(x) = r(t)c

· cs(x), with c arbitrary non-zero constant, we impose a

fixing condition, say

s(X0) = S0. (7.6)

In the above setting, the functionsu0, χ, β and the constantS0 are given. We further

assume that the conditions (7.2)–(7.5) are consistent, i.e. the following compatibility

conditions are satisfied:

u′0(0) = u′0(L) = β ′(0) = β ′(L) = 0, χ(0) = u0(X0), χ(T ) = β(X0). (7.7)

The unique solvability, i.e. existence and uniqueness of the solution of the inverse

problem (7.1)–(7.6), was established in [47]. With some slight corrections, this theo-

rem reads as follows.

Theorem 7.2.1 Suppose thatu0(x), β(x) ∈ W42(0, L), andχ(t) ∈ W

22(0, T ) satisfy

(7.7)and thatS0 6= 0. Also, assume that:

(i) M := χ′(0)− u′′0(X0) 6= 0, m :=χ′(T )− β ′′(X0)

M6= 0,

(ii) u′′′0 (0) = u′′′0 (L) = β ′′′(0) = β ′′′(L) = 0,

(iii) λ1 < 1, 4λ2λ3 − (1− λ1)2 ≤ 0, λ4 < 1,

where

λ1 :=2

m2M2max

M2 +4L2‖χ′′‖2

π2, 4L‖θ‖2 + Lm2‖u′′′0 ‖2

,

λ2 :=2

M2max

4L6

π4m4, 1

, λ3 :=2‖θ′‖2m2

+4‖χ′′‖2M2

(

2‖θ‖2m2

+ ‖u′′′0 ‖2)

,

λ4 :=1

m2M2max M2 + 2L3z + 4L2‖χ′′‖2, 4L3z + 4L3‖θ′‖2 + 2Lm2‖u′′′0 ‖2,

θ(x) = β ′′′(x)−mu′′′0 (x), z =1− λ12λ2

.

Then the inverse problem given by equations(7.1)–(7.6)has a unique solutionr(t) ∈W

12(0, T ), s(x) ∈ W

22(0, L) and

u(x, t) ∈ W4,22 (DT ) ∩ C(0, T ;W4

2(0, L)) ∩ C(0, L;W22(0, T )).

Chapter 7. 152

Note that in the above theorem,Wk2(Ω), with k ∈ 1, 2, 4 andΩ = (0, L) or

(0, T ), denotes the Sobolev space of functions consisting of all elements ofL2(Ω)

having generalised derivatives up to orderk inclusively inL2(Ω). Also, we denote

W4,22 (DT ) := u ∈ L2(DT )|∂jxu ∈ L2(0, L), j = 1, 4, and∂itu ∈ L2(0, T ), i = 1, 2.

Finally, C(0, T ;W42(0, L)) denotes the space of continuous mappings from(0, T ) to

W42(0, L) andC(0, L;W2

2(0, T )) denotes the space of continuous mappings from(0, L)

to W22(0, T ). The norms‖χ′′‖ and‖u′′′0 ‖ are understood inL2(0, T ) andL2(0, L),

respectively. Also, the norms ofθ andθ′ are inL2(0, L).

Although the inverse problem (7.1)–(7.6) has a unique solution it is still ill-posed

because it violates the continuous dependence upon the input data (7.4) and (7.5). In

the next section we will demonstrate how the BEM discretising numerically the heat

equation (7.1) can be used together with the regularisationin order to obtain a stable

solution.


In this section, we use the numerical procedure for discretising the inverse problem

(7.1)–(7.6) by using the BEM which results in the following boundary integral equa-

tion:

η(x)u(x, t) =

∫ t

0

[

G(x, t, ξ, τ)∂u

∂n(ξ)(ξ, τ)− u(ξ, τ)

∂G

∂n(ξ)(x, t, ξ, τ)

]

ξ∈0,L

dτ

+

∫ L

0

G(x, t, y, 0)u(y, 0) dy+

∫ L

0

∫ T

0

G(x, t, y, τ)r(τ)s(y) dτdy,

(x, t) ∈ [0, L]× (0, T ]. (7.8)

Chapter 7. 153

Using the same discretisation as described in Section 2.3 which has been used so far

as in previous chapters, we obtain

η(x)u(x, t) =N∑

j=1


+

N0∑

k=1

[Ck(x, t)u0,k +Drk(x, t)sk] , (7.9)

where

Drk(x, t) =

∫ xk

xk−1

∫ t

0

G(x, t, y, τ)r(τ) dτ dy =

N∑

j=1

dj,k(x, t)rj, (7.10)

wheredj,k(x, t) =∫ xkxk−1

∫ tjtj−1

G(x, t, y, τ) dτ dy for j = 1, N , k = 1, N0. The double

integral source termdj,k(x, t) can be evaluated analytically to be given by

dj,k(x, t) =

0 ; t ≤ tj−1,

J(x, t, xk−1, tj−1)− J(x, t, xk, tj−1)

+(x− xk−1)

2

4− (x− xk)

2

4; tj−1 < t ≤ tj , x ≤ xk−1,

J(x, t, xk−1, tj−1)− J(x, t, xk, tj−1)

−(x− xk−1)2

4− (x− xk)

2

4; tj−1 < t ≤ tj , xk−1 < x ≤ xk,

J(x, t, xk−1, tj−1)− J(x, t, xk, tj−1)

−(x− xk−1)2

4+

(x− xk)2

4; tj−1 < t ≤ tj , x > xk,

J(x, t, xk−1, tj−1)− J(x, t, xk, tj−1)

−J(x, t, xk−1, tj) + J(x, t, xk, tj) ; t > tj ,

Chapter 7. 154

where

J(x, t, xk, tj) =

(

(x− xk)2

4+t− tj2

)

erf

(

x− xk2√t− tj

)

+

√t− tj2√π

(x− xk) exp

(

−(x− xk)2

4(t− tj)

)

.

By applying (7.9) at the boundary element nodes(0, ti) and(L, ti) for i = 1, N and

the homogeneous Neumann boundary condition (7.3), i.e.q0j = qLj = 0, we obtain

the system of2N equations

−Bh¯+ Cu

¯0+Drs

¯= 0

¯, (7.11)

whereDr =

∑Nj=1 dj,k(0, ti)rj

∑Nj=1 dj,k(L, ti)rj

2N×N0

.

For the direct problem, we can find now the boundary temperaturesu(0, ti) and

u(L, ti) from (7.11) as

h¯= B−1(Cu

¯0+Drs

¯). (7.12)

Furthermore, the interior temperaturesu(X0, ti) for i = 1, N from the additional con-

dition condition (7.4) can be approximated similarly as in (6.16), i.e. [u(X0, ti)]N =[

χ(ti)]

N. Applying this in (7.9) it gives

−BIh¯+ CIu

¯0+DrIs

¯= χ

¯, (7.13)

whereDrI =[

∑Nj=1 dj,k(X0, ti)rj

]

N×N0

. Whereas the final temperatureu(xk, T ) for

k = 1, N0 from the overdetermination (7.5) can be approximated as[u(xk, T )]N0=

[

β(xk)]

N0

. Applying this in (7.9) it gives

−BIIIh¯+ CIIIu

¯0+DrIIIs

¯= β

¯, (7.14)

Chapter 7. 155

where

BIII =[

B0j(xk, T ) BLj(xk, T )]

N0×2N, CIII =

[

Ck(xk, T )]

N0×N0

,

DrIII =[

∑Nj=1 dj,k(xk, T )rj

]

N0×N0

.

7.4 Solution of inverse problem

In this section, we wish to obtain simultaneously the unknown componentsr(t) and

s(x) of the multiplicative source term in the inverse problem (7.1)–(7.6) by using the

BEM together with a classical minimisation process. The conditions (7.4)–(7.6) are

imposed by minimising the nonlinear least-squares function

F0(r, s) :=

N∑

i=1

(

u(X0, ti)− χ(ti))2

+

N0∑

k=1

(u(xk, T )− β(xk))2 + (s(X0)− S0)

2.

(7.15)

Here, the approximated temperaturesu(X0, ti) andu(xk, T ), as introduced earlier in

(7.13) and (7.14), respectively, are now employed into the above objective function

with the initial guesses r¯0

and s¯0

for functionsr ands, respectively. Whereass(X0) is

approximated the same as in (6.20). Then, applying the approximations (7.12)–(7.14)

we obtain

F0(r¯, s¯) =‖ − BIB−1(Cu

¯0+Drs

¯) + CIu

¯0+DrIs

¯− χ

¯‖2

+ ‖ − BIIIB−1(Cu¯0

+Drs¯) + CIIIu

¯0+DrIIIs

¯− β

¯‖2

+ (s(X0)− S0)2, (7.16)

where r¯= (rj)N and s

¯= (sk)N0

. The minimisation of (7.16) is performed using the

lsqnonlinroutine from the MATLAB Optimisation Toolbox. This routineattempts to

find the minimum of a sum of squares by starting from some arbitrary initial guesses r¯0

and s¯0

. Note that we have compiled this routine with the following default parameters:

Chapter 7. 156

• Algorithm = Trust-Region-Reflective.

• Maximum number of objective function evaluations, ‘MaxFunEvals’ = 100 ×(N +N0 + 1).

• Maximum number of iterations, ‘MaxIter’ =400.

• Termination tolerance on the function value, ‘TolFun’ =10−10 to 10−6.

• Termination tolerance, ‘TolX’ =10−10 to 10−6.

Of course, finding a global minimiser to a nonlinear optimisation problem is not an

easy task since the functional (7.15), which is in general not convex, i.e. the Hessian

of F is not positive definite. As a consequence it may have local minima in which a

descent method tends to get stuck, if the underlying inverseproblem is ill-posed, [14,

p.17]. In the next section we shall elaborate more on the choice of the initial guess for

the iterative routine, as well as on incorporating regularisation in the functional (7.15)

in order to ensure convergence to the desired stable solution.


This section presents three benchmark test examples in order to test the accuracy and

stability of the numerical methods introduced in Sections 7.3 and 7.4. The RMSEs

for r(t) ands(x), defined in (2.49) and (6.28), respectively, are used to evaluate the

accuracy of the numerical results.

7.5.1 Example 1

We consider a benchmark test example withT = 1, L = 1/10, X0 = 1/20, and the

initial data (7.2) given by

u0(x) = u(x, 0) = 0, x ∈ [0, L]. (7.17)

Chapter 7. 157

For the direct problem (7.1)–(7.3) we also need the input source data

r(t) = − et

40

(

400π2t2 − 400π2t + t2 + t− 1)

, s(x) = 40 cos(20πx). (7.18)

In order to test the BEM accuracy for the direct problem givenby equation (7.1) with

the source given by the product of the functions in (7.18), subject to the homogenous

Neumann boundary condition (7.3) and the initial condition(7.17), the numerical re-

sults are compared with the analytical solution given by

u(x, t) = et(t− t2) cos(20πx). (7.19)

The exact expressions for the inputs (7.4)–(7.6) are given by

χ(t) = u(1/20, t) = −(t− t2)et, β(x) = u(x, 1) = 0,

S0 = s(1/20) = −40.

(7.20)

As defined in Theorem 7.2.1, we then haveS0 = −40 6= 0,M = −1 6= 0,m = −e 6=0, θ(x) = β(x) = u0(x) ≡ 0, λ1 = 0.2962 < 1, λ2 = 2, λ3 = 0, 4λ2λ3 − (1− λ1)

2 =

−0.4953 ≤ 0, z = 0.1759, andλ4 = 0.2613 < 1 which satisfy all the conditions

(i)–(iii) for existence and uniqueness of the solution.

As the specified boundary conditions (7.3) are of Neumann type, the boundary

unknowns in the BEM are represented by the Dirichlet datau(0, t) andu(L, t), as given

by (7.12). Once all the boundary data has been obtained accurately, equations (7.13)

and (7.14) can be employed explicitly and with no need of interpolations to obtain

the temperaturesu( 120, ti) andu(xk, 1) for i = 1, N andk = 1, N0, respectively. The

RMSE of the direct problem results are shown in Table 7.1 and it can be concluded that

the BEM numerical solutions are convergent to their corresponding exact values, as the

number of boundary elements increases. Whereas Figure 7.1 displays the analytical

and numerical results ofχ(t) andβ(x) and very good agreement can be observed.

Next we consider the inverse problem given by equations (7.1), (7.3), (7.17) and

Chapter 7. 158

Table 7.1: The RMSE foru(0, t), u(0.1, t), χ(t) andβ(x), obtained using the BEM forthe direct problem withN = N0 ∈ 10, 20, 40, for Example 1.

N = N0RMSE

u(0, t) u(0.1, t) χ(t) β(x)10 5.01E-3 5.01E-3 5.64E-3 8.51E-220 1.03E-3 1.03E-3 1.75E-3 4.51E-240 8.17E-4 8.17E-4 9.69E-4 2.30E-2

0 0.2 0.4 0.6 0.8 1−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

t

χ(t

)

(a)

0 0.02 0.04 0.06 0.08 0.1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x

β(x

)

(b)

Figure 7.1: The analytical (—–) and numerical results for (a) χ(t) and (b) β(x)obtained using the BEM for the direct problem withN = N0 ∈ 10 (− ·−), 20 (· · · ), 40 (−−−), for Example 1.

(7.20). The numerical solution can be obtained, as described in Section 7.4, by min-

imising the objective function (7.15). Preliminary numerical investigations showed

that the initial guesses r¯0

and s¯0

cannot be so arbitrary in order for the minimisation

process to converge globally. After many trials, we decidedto illustrate the numerical

results obtained by considering the initial guess as

r¯0

= r¯+ random(′Normal′, 0, σr, N, 1),

s¯0

= s¯+ random(′Normal′, 0, σs, N0, 1),

(7.21)

with the standard deviationsσr andσs, respectively, given by

σr = p0 × maxt∈[0,T ]

|r(t)|, σs = p0 × maxx∈[0,L]

|s(x)|, (7.22)

Chapter 7. 159

wherep0 is a percentage of perturbation. Hereafter, unless otherwise specified, we

present results obtained withp0 = 100% perturbed initial guess (which is quite far

from the exact solution (7.18)) andN = N0 = 20, we also have set parameters TolFun

= TolX = 10−6 for the MATLAB optimisation toolboxlsqnonlinto solve the inverse

problem.

Figure 7.2(a) shows the unregularised objective functionF0 which converges in

39 iterations and the numerical results forr(t), s(x), u(0, t), u(0.1, t) are displayed

in Figures 7.2(b)–7.2(e), respectively. As we can see in these figures, the numerical

results are inaccurate and partially unstable in Figure 7.2(c).

In order to improve the accuracy and stability, we apply a Tikhonov regularisation

process based on minimising the penalised objective function

Fλ(r¯, s¯) := F0(r

¯, s¯) + λ

(

‖Rr‖2 + ‖Rs‖2)

, (7.23)

whereλ > 0 is a regularisation parameter to be prescribed, andR is a (differential)

regularising matrix as introduced in Section 1.6. Initially, we have applied the first-

and second-order regularisations based on minimising the objective function (7.23) as

Fλ(r¯, s¯) =F0(r

¯, s¯) + λ

(

N−1∑

i=1

(ri+1 − ri)2 +

N0−1∑

k=1

(sk+1 − sk)2

)

, (7.24)

Fλ(r¯, s¯) =F0(r

¯, s¯) + λ

(

N−1∑

i=2

(−ri+1 + 2ri − ri−1)2 +

N0−1∑

k=2

(−sk+1 + 2sk − sk−1)2

)

,

(7.25)

respectively.

By trial and error, among various regularisation parameters λ ∈ 10−9, . . . , 102,

we have found, as illustrative stable results, those obtained withλopt = 10−5 which

are shown in Figure 7.3. As we can see in this figure, applying orders one or two

regularisations (7.24) or (7.25) yield stable, but rather inaccurate results, especially

near the endpoints of the intervals of definition of the functions involved, see Figure

Chapter 7. 160

0 10 20 30 4010

−4

10−2

100

102

104

106

108

Number of iterations

Obje

ctiv

eFunct

ion

F0

39

(a)

0 0.2 0.4 0.6 0.8 1−100

−50

0

50

100

150

t

r(t

)

(b)

0 0.02 0.04 0.06 0.08 0.1−150

−100

−50

0

50

100

150

x

s(x

)

(c)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

u(0

,t)

(d)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

u(0

.1,t)

(e)

Figure 7.2: (a) The objective functionF0 and the numerical results for (b)r(t), (c)s(x), (d) u(0, t), (e) u(0.1, t) obtained with no regularisation(− · −), for exact datafor Example 1. The corresponding analytical solutions are shown by continuous line(—–) in (b)–(e) and thep0 = 100% perturbed initial guesses are shown by dotted line(· · · ) in (b) and (c).

Chapter 7. 161

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

t

r(t

)

(a)

0 0.02 0.04 0.06 0.08 0.1−60

−40

−20

0

20

40

60

x

s(x

)

(b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

u(0

,t)

(c)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

u(0

.1,t

)

(d)

Figure 7.3: The numerical results for (a)r(t), (b)s(x), (c)u(0, t), (d)u(0.1, t) obtainedwith the first-order regularisation(· · · ) and the second-order regularisation(− − −)with regularisation parameterλopt = 10−5, for exact data for Example 1. The corre-sponding analytical solutions are shown by continuous line(—–).

7.3(b). In order to improve on these inaccuracies we have then investigated a hybrid

combination of first- and second- order regularisations given by

Fλ(r¯, s¯) = F0(r

¯, s¯) + λ

(

(r1 − r2)2 + (−rN−1 + rN)

2 +

N−1∑

i=2

(−ri+1 + 2ri − ri−1)2

+ (s1 − s2)2 + (−sN0−1 + sN0

)2 +

N0−1∑

k=2

(−sk+1 + 2sk − sk−1)2

)

. (7.26)

Chapter 7. 162

According to (7.23) and (7.26), the differential regularisation matrixR is given by

R =

1 −1 0 0 .

−1 2 −1 0 .

0 −1 2 −1 .

. . . . .

. (7.27)

In the regularisation process, we need to choose an appropriate regularisation parame-

terλ which balances accuracy and stability. Here, we use theL-curve method to find

the regularisation parameterλ. Figure 7.4(a) shows theL-curve obtained by plotting

the solution norm√

‖Rr‖2 + ‖Rs‖2 versus the residual norm√

F0(r¯, s¯) for various

values ofλ whenR is given by (7.27). From this figure it can be seen that the corner of

theL-curve occurs nearbyλL = 10−5, with other appropriate values between the wide

range10−6 to 10−4. With this value of the regularisation parameter, the regularised ob-

jective functionFλ and the numerical results are shown in Figures 7.4(b)–7.4(f). From

Figure 7.4(b) it can be seen that convergence for the regularised objective functionFλ

is achieved within 15 iterations. Also, in comparison with the previous Figures 7.3(a)–

7.3(d), very good agreement between the exact and the regularised numerical solutions

is now obtained, as illustrated in Figures 7.4(c)–7.4(f), respectively. All results are

summarised in terms of the RMSE in Table 7.2. Various initialguesses (7.21) with

p0 ∈ 40, 60, 80, 100% in (7.22) have been investigated in order to test the robustness

of the minimisation procedure with respect to the independence on the initial guess.

From Table 7.2 it can be seen that whilst the choice of the initial guess seems to matter

for the accuracy of the unregularised solution; i.e.λ = 0, this restriction disappears

when regularisation withλL = 10−5 is imposed. This shows that the numerical reg-

ularisation method employed is robust with respect to the independence on the initial

guess.

To test the stability of the BEM combined with the nonlinear regularisation, we

solve the inverse problem when random noises are added to theinput functionsχ(t)

Chapter 7. 163

10−2

10−1

100

101

0

20

40

60

80

100

120λ=10−9

λ=10−8

λ=10−7

λ=10−6

λ=10−5

λ=10−4λ=10−3

λ=10−2λ=10−1 λ=1

√

F0(r¯, s¯)

√

‖R

r ¯‖2

+‖R

s ¯‖2

(a)

0 5 10 1510

−4

10−2

100

102

104

106

108


Obje

ctiv

eFunct

ion

Fλ

15

(b)

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

t

r(t

)

(c)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09−50

−40

−30

−20

−10

0

10

20

30

40

50

x

s(x

)

(d)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

u(0

,t)

(e)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

u(0

.1,t)

(f)

Figure 7.4: (a) TheL-curve criterion, (b) the objective functionFλ, and the numericalresults(− −) for (c) r(t), (d) s(x), (e)u(0, t), (f) u(0.1, t) obtained with the hybrid-order regularisation (7.26) with regularisation parameter λL = 10−5 suggested byL-curve, for exact data Example 1. The corresponding analytical solutions are shown bycontinuous line (—–) in (c)–(f).

Chapter 7. 164

0 2 4 6 8 10 12 14 1610

−4

10−2

100

102

104

106

108


Obje

ctiv

eFunct

ion

Fλ

16

15

15

(a)

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

t

r(t

)

(b)

0 0.02 0.04 0.06 0.08 0.1−50

−40

−30

−20

−10

0

10

20

30

40

50

x

s(x

)

(c)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

u(0

,t)

(d)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

u(0

.1,t

)

(e)

Figure 7.5: (a) The objective functionFλ and the numerical results for (b)r(t), (c)s(x),(d) u(0, t), (e)u(0.1, t) obtained with the hybrid-order regularisation (7.26) withregu-larisation parameterλL = 10−5 suggested byL-curve forp ∈ 1(−·−), 3(· · · ), 5(−−−)% noisy data, for Example 1. The corresponding analytical solutions are shown bycontinuous line (—–) in (b)–(e).

Chapter 7. 165

Table 7.2: The RMSE forr(t), s(x), u(0, t), u(0.1, t) for exact data, Example 1.

p0(%)parameter RMSE

λ r(t) s(x) u(0, t) u(0.1, t)

40%0 6.349 18.47 4.72E-2 3.27E-2

λL=1E-5 1.528 0.819 1.49E-2 1.53E-2

60%0 9.752 26.70 1.14E-1 8.12E-2

λL=1E-5 1.513 0.767 1.48E-2 1.50E-2

80%0 25.83 44.93 1.99E-1 2.74E-1

λL=1E-5 1.526 0.812 1.48E-2 1.47E-2

100%0 53.70 54.24 2.34E-1 2.54E-1

λL=1E-5 1.529 0.819 1.47E-2 1.48E-2

andβ(x) as

χ¯ǫ = χ

¯+ random(′Normal′, 0, σχ, N, 1),

β¯ǫ = β

¯+ random(′Normal′, 0, σβ, N0, 1),

(7.28)

with the standard deviationsσχ andσβ given by

σχ = p× maxt∈[0,T ]

|χ(t)|, σβ = p× maxx∈[0,L]

|β(x)|, (7.29)

respectively. The numerical results obtained withλL = 10−5, are illustrated in Fig-

ure 7.5. From Figure 7.5(a) it can be seen that convergence ofthe hybrid-order reg-

ularised objective functional (7.26) is rapidly achieved within 15-16 iterations for

p ∈ 1, 3, 5%. Furthermore, Figures 7.5(b)–7.5(e) show that stable and accurate

numerical results are obtained for all amounts of noisep. Also, as expected, the nu-

merical solutions become more accurate as the amount of noisep decreases.

7.5.2 Example 2

In Example 1, all conditions for the existence and uniqueness of Theorem 7.2.1 were

satisfied. We now consider an example which has the analytical solution, [48],

u(x, t) = (e3t − e−t) cos(x), r(t) = e3t, s(x) = 4 cos(x), (7.30)

Chapter 7. 166

whereT = 0.3, L = π. One can easily check that the homogeneous Neumann con-

ditions (7.3) are satisfied and that the initial condition (7.2) is also homogeneous, as

given in (7.17). TakingX0 = 0.75 we obtain that the input data (7.4)–(7.6) are given

by

χ(t) = u(0.75, t) = (e3t − e−t) cos(0.75),

β(x) = u(x, 0.3) = (e0.9 − e−0.3) cos(x),

S0 = s(0.75) = 4 cos(0.75).

(7.31)

From this we haveS0 = M = 4 cos(0.75) 6= 0, u0(x) ≡ 0, m = e0.9 6= 0, θ(x) =

(e0.9 − e−0.3) sin(x), λ1 = 5.3719, λ2 = 0.2518, λ3 = 24.928, z = −8.6813, λ4 =

14.654. One can then observe that the conditions (i) and (ii) of Theorem 7.2.1 are

satisfied, but the condition (iii) has been violated. Whilsta solution obviously exists,

as given by equations (7.30), one cannot guarantee yet that this solution is unique.

We have solved first the direct problem given by equations (7.1) (with r ands given

by (7.30)), (7.3) and (7.17) using the BEM with various numbers of boundary elements

N = N0 ∈ 5, 10, 20 and the numerical results forχ(t) andβ(x) presented in Fig-

ure 7.6 show rapid convergence and excellent agreement withthe analytical solution

(7.31). Afterwards, we have solved the inverse problem given by equations (7.1), (7.3),

(7.17) and (7.31) in order to retrieve the temperatureu(x, t) and the heat source com-

ponentsr(t) ands(x) given analytically by (7.30). We have taken boundary elements

N = N0 = 20 and the arbitrary initial guesses r¯0

= 0¯

and s¯0

= 0¯.

We first consider the case of exact data. The convergence of the unregularised

objective functionF0 achieved within 56 iterations using thelsqnonlin routine with

TolFun = TolX =10−10 is illustrated in Figure 7.7(a). Also, the RMSEs of solutions r

ands are shown in Figure 7.7(b) by dash line(· · · ) and dot line(−−−), respectively.

The numerical solutions forr ands obtained after 56 iterations are shown by dash-dot

line (− · −) in Figures 7.7(c) and 7.7(d), respectively. Very good agreement between

the numerical and analytical solutions fors can be observed, whilst the numerical

solution forr is stable but slightly away from the analytical solution. Wethen look

Chapter 7. 167

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.2

0.4

0.6

0.8

1

1.2

1.4

t

χ(t

)

(a)

0 0.5 1 1.5 2 2.5 3 3.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

β(x

)

(b)

Figure 7.6: The analytical (—–) and numerical results for (a) χ(t) and (b)β(x)obtained using the BEM for the direct problem withN = N0 ∈ 5(− ·−), 10(· · · ), 20(−−−), for Example 2.

more closely at Figure 7.7(b) and observe that the minimum ofRMSEs is at iteration

31 instead of 56. Therefore, we have tried solving the inverse problem with the fixed

iteration at 31, and the numerical results become more accurate, as illustrated by the

circle markers() in Figures 7.7(c) and 7.7(d). Further, we have applied the hybrid-

order regularisation procedure (7.26) with the regularisation parameterλopt = 2 ×10−4 (chosen by the trial and error) and the results are shown in Figure 7.8. Figure

7.8(a) displays the convergence of the regularised functional (7.26) achieved within

28 iterations. Also, results for RMSEs and the solutions forr and s are shown in

Table 7.3: The RMSE forr(t) ands(x), for the noise levelsǫ0 ∈ 0, 0.01, 0.1, forExample 2.

Noise level No. of iterationsparameter RMSE

λ r(t) s(x)

No noise

56 0 6.3068E-2 1.5492E-131 (fixed) 0 2.6668E-2 1.4751E-1

28 λopt=2E-4 5.6471E-2 1.1003E-123 (fixed) λopt=2E-4 1.9713E-2 6.4412E-2

ǫ0 = 0.0127 λopt=4E-4 6.3004E-2 1.0886E-1

21 (fixed) λopt=4E-4 2.8829E-2 6.5281E-2ǫ0 = 0.1 17 λopt=2 5.2212E-2 3.0714E-2

Chapter 7. 168

Figures 7.8(b)–7.8(d). From Figure 7.8(b) one can see that the minimum of the RMSEs

occurs after 23 iterations. By comparing Figures 7.7 and 7.8one can conclude that the

inclusion of some small regularisation yields slightly more accurate and stable results.

Next, we consider the stability of the numerical solution when the noise is present

0 10 20 30 40 50 6010

−10

10−8

10−6

10−4

10−2

100

102


Obje

ctiv

eFunct

ion

F0

56

31

(a)

0 10 20 30 40 50 6010

−2

10−1

100

101


Root

Mea

nSquare

Err

or

(b)

0 0.05 0.1 0.15 0.2 0.25 0.30.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

t

r(t

)

(c)

0 0.5 1 1.5 2 2.5 3 3.5−5

−4

−3

−2

−1

0

1

2

3

4

5

x

s(x

)

(d)

Figure 7.7: (a) The objective functionF0, (b) the RMSEs forr(t) (− − −) ands(x)(· · · ) obtained with no regularisation for exact data, and the numerical results for (c)r(t) and (d)s(x) obtained using the minimisation process after 56 unfixed iterations(− · −), and 31 fixed iterations( ), for Example 2. The corresponding analyticalsolutions (7.30) are shown by continuous line (—–) in (c) and(d).

Chapter 7. 169

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

100

101

102


Obje

ctiv

eFunct

ion

Fλ

2823

(a)

0 5 10 15 20 25 3010

−2

10−1

100

101


Root

Mea

nSquare

Err

or

(b)

0 0.05 0.1 0.15 0.2 0.25 0.30.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

t

r(t

)

(c)

0 0.5 1 1.5 2 2.5 3 3.5−5

−4

−3

−2

−1

0

1

2

3

4

5

x

s(x

)

(d)

Figure 7.8: (a) The objective functionFλ, (b) the RMSEs forr(t) (− − −) ands(x)(· · · ) obtained using the hybrid-order regularisation (7.26) with regularisation param-eterλopt = 2 × 10−4 for exact data, and the numerical results for (c)r(t) and (d)s(x)obtained using minimisation process after 28 unfixed iterations(− · −), and 23 fixediterations( ), for Example 2. The corresponding analytical solutions (7.30) areshown by continuous line (—–) in (c) and (d).

in the input data (7.4) and (7.5). As in [48], the noise was defined by

χǫ(ti) = χ(ti)

1 +ǫ0

√

∑Ni=1 χ

2(ti)rand(i)

, i = 1, N,

βǫ(xk) = β(xk)

1 +ǫ0

√

∑Nk=1 β

2(xk)rand(k)

, k = 1, N0,

(7.32)

whererand(·) is a random variable generated by the MATLAB command from a nor-

Chapter 7. 170

0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

101

102


Obje

ctiv

eFunct

ion

Fλ

2127

(a)

0 5 10 15 20 25 3010

−2

10−1

100

101


Root

Mea

nSquare

Err

or

(b)

0 0.05 0.1 0.15 0.2 0.25 0.30.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

t

r(t

)

(c)

0 0.5 1 1.5 2 2.5 3 3.5−5

−4

−3

−2

−1

0

1

2

3

4

5

x

s(x

)

(d)

Figure 7.9: (a) The objective functionFλ, (b) the RMSEs forr(t) (− − −) ands(x)(· · · ) obtained using the hybrid-order regularisation (7.26) with regularisation parame-terλopt = 4× 10−4 for noise levelǫ0 = 0.01, and the numerical results for (c)r(t) and(d) s(x) obtained using the minimisation process after 27 unfixed iterations(− · −),and 21 fixed iterations( ), for Example 2. The corresponding analytical solutions(7.30) are shown by continuous line (—–) in (c) and (d).

mal distribution with mean zero and unit standard deviation, and ǫ0 represents the

noise level. Remark that the noise (7.32) is multiplicative, whilst the noise in (7.28),

Example 1, is additive. Forǫ0 = 0.01, Figure 7.9 illustrates the results obtained using

the hybrid-order regularisation (7.26) with regularisation parameterλopt = 4 × 10−4.

The convergence of the regularised objective function achieved within 27 iterations is

shown in Figure 7.9(a), whilst the minimum RMSEs ofr ands occur after 21 itera-

tions, as can be seen in Figure 7.9(b). Numerical solutions for r ands obtained after

Chapter 7. 171

0 2 4 6 8 10 12 14 16 1810

−1

100

101

102


Obje

ctiv

eFunct

ion

Fλ

(a)

0 5 10 15 2010

−2

10−1

100

101


Root

Mea

nSquare

Err

or

(b)

0 0.05 0.1 0.15 0.2 0.25 0.31

1.5

2

2.5

t

r(t

)

(c)

0 0.5 1 1.5 2 2.5 3 3.5−4

−3

−2

−1

0

1

2

3

4

x

s(x

)

(d)

Figure 7.10: (a) The objective functionFλ, (b) the RMSEs forr(t) (− − −) ands(x) (· · · ) obtained using the hybrid-order regularisation (7.26) with regularisationparameterλopt = 2 for noise levelǫ0 = 0.1, and the numerical results(− · −) for (c)r(t) and (d)s(x) obtained using the minimisation process after 17 (unfixed) iterations,for Example 2. The corresponding analytical solutions (7.30) are shown by continuousline (—–) in (c) and (d).

27 (unfixed) and 21 (fixed) iterations are displayed in Figures 7.9(c) and 7.9(d), re-

spectively. As expected, the conclusions from Figure 7.9 obtained for a low level of

noiseǫ0 = 0.01 are very much the same as the those from Figure 7.8 obtained for no

noiseǫ0 = 0. From both Figures 7.8(c), 7.8(d) and 7.9(c), 7.9(d) one canobserve that

the numerical results are accurate and stable. Furthermore, there is little difference in

the results obtained whether we stop (fix) the iteration process at the minimum of the

RMSEs shown in Figures 7.8(b) and 7.9(b) or, if we let the iteration process running

Chapter 7. 172

(unfix) until converge of the regularised objective function is achieved.

Next, we consider a large amount of noise, such asǫ0 = 0.1, included in (7.32)

and the numerical results are shown in Figure 7.10. First, one can observe from Figure

7.10(a) that the convergence of the objective function (7.26) is rapidly achieved within

17 iterations and the monotonic decreasing curve has a somewhat different shape than

that recorded in Figure 7.8(a) for no noiseǫ0 = 0 or in Figure 7.9(a) for a low amount

noiseǫ0 = 0.01. Also, interestingly, unlike in Figures 7.8(b) and 7.9(b) where the

RMSEs show a minimum before the iteration process has finished, in Figure 7.10(b)

no such minimum occurs. Therefore, in Figures 7.10(c) and 7.10(d) we present only

numerical results forr ands, respectively, obtained after 17 (unfixed) iterations with

λopt = 2. From these figures it can be seen that the numerical solutions are stable, with

an unexpected very high accuracy in predicting thes component in Figure 7.10(d). For

completeness and clarity the RMSEs of Figures 7.7(b)–7.10(b) are given in numbers

in Table 7.3. From this table, and also from Figure 7.10(b), it can be seen that for

ǫ0 = 0.1 the components(x) is predicted more accurately than ther(t) component,

whilst the prediction forǫ0 ∈ 0, 0.01 is reversed.

Finally, we report that the numerical results presented in this example are compa-

rable in terms of accuracy and stability with the numerical results obtained recently

in [48] using a different method of successive approximantspreviously developed in

[47].

7.5.3 Example 3

The previous examples possessed an analytical (smooth) solution available explicitly

and they were tested in order to verify the accuracy and stability of the numerical

method employed. In this subsection, we consider a severe test example represented

Chapter 7. 173

by the non-smooth source components

r(t) =

t, 0 ≤ t ≤ 1/2,

1− t, 1/2 < t ≤ 1 = T,

(7.33)

s(x) =

x, 0 ≤ x ≤ 1/20,

0.1− x, 1/20 < x ≤ 1/10 = L,

(7.34)

whereL = 1/10, T = 1, X0 = 1/20. We also take the homogeneous initial tem-

perature (7.17). This example does not have an analytical solution for the temperature

u(x, t) readily available. Therefore, in such a situation the data (7.4) and (7.5) is sim-

ulated numerically by solving the direct problem (7.1) withthe multiplicative source

given by the product of the functions in (7.33) and (7.34), subject to the homogeneous

boundary and initial conditions (7.3) and (7.17). The BEM numerical solutions for the

dataχ(t) = u(0.2, t) andβ(x) = u(x, 1) are shown in Figure 7.11 for various numbers

of boundary elementsN = N0 ∈ 10, 20, 40. From this figure the convergence of the

numerical solution, as the number of boundary elements increases, can be observed.

Next we consider the inverse problem given by equations (7.1), (7.3), (7.6) with

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7x 10

−3

t

χ(t

)

(a)

0 0.02 0.04 0.06 0.08 0.16.3

6.35

6.4

6.45

6.5

6.55

6.6x 10

−3

x

β(x

)

(b)

Figure 7.11: The numerical results for (a)χ(t) and (b)β(x) obtained using the BEMfor the direct problem withN = N0 ∈ 10(−·−), 20(· · · ), 40(−−−), for Example3.

Chapter 7. 174

0 1 2 3 4 5 6 7 810

−7

10−6

10−5

10−4

10−3

10−2


Obje

ctiv

eFunct

ion

Fλ

(a)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

r(t

)

(b)

0 0.02 0.04 0.06 0.08 0.10

0.01

0.02

0.03

0.04

0.05

0.06

x

s(x

)

(c)

Figure 7.12: (a) The objective functionFλ and the numerical results for (b)r(t) and (c)s(x) obtained with the hybrid-order regularisation (7.26) withregularisation parameterλ = 2 × 10−4 for p ∈ 1(− · −), 5(· · · ), 10(− − −)% noisy data for Example 3.The corresponding analytical solutions (7.33) and (7.34) are shown by continuous line(—–) in (b) and (c).

S0 = s(1/20) = 1/20 specified, (7.17), and the additional measured data (7.4) and

(7.5) which has been simulated numerically in Figure 7.11. We pick from Figure 7.11

the numerical BEM solutions obtained withN = N0 = 20 and we further perturb this

data with noise, as in (7.28). We took as initial guesses r¯0

= s¯0

= 0, and we initiated

the iterative minimisation process of the hybrid-order regularisation functional (7.26),

as described in Example 1. The numerical results obtained withλopt = 2×10−4 (found

by trial and error) are shown in Figure 7.12 forp ∈ 1, 5, 10% noise generated as in

(7.28). From Figure 7.12(a) it can be seen that the convergence of the functional (7.26)

Chapter 7. 175

is rapidly achieved within 7-8 iterations using thelsqnonlinroutine with TolFun = TolX

= 10−6. Also, Figures 7.12(b) and 7.12(c) show that stable and accurate numerical

solutions for bothr(t) ands(x) are obtained for all the amounts of noisep.

In closure, although not illustrated, we report that the same good performance has

been recorded when attempting to reconstruct even discontinuous source components.

7.6 Conclusions

In this chapter, inverse source problems with homogeneous Neumann boundary con-

ditions together with specified interior and final time temperature measurements have

been considered to find the space- and the time-dependent components of a multiplica-

tive source function. The numerical discretisation was based on the BEM combined

with a nonlinear Tikhonov regularisation procedure via thelsqnonlinroutine from the

MATLAB. For a wide range of test examples, the obtained results indicate that stable

and accurate numerical solutions have been achieved. The identification of both multi-

plicativer(t)s(x) and additiver1(t)+s1(x) components of space- and time-dependent

sources of the formr(t)s(x) + r1(t) + s1(x) can also be considered, [47], but its nu-

merical implementation is deferred to a future work.

Chapter 8. 176

Chapter 8

General Conclusions and Future Work

8.1 Conclusions

The aim of this thesis was to solve various inverse source problems for the (one-

dimensional) heat equation by using the BEM to find the time-dependent heat source

function, presented in Chapters 2–5, and space- and time-dependent heat source func-

tions for additive and multiplicative cases, presented in Chapters 6 and 7, respectively.

Several types of conditions such as non-local, non-classical, periodic, fixed point, time-

average and integral have been considered as boundary or overdetermination condi-

tions.

The BEM has been used as the main numerical approach for discretising the linear

heat equation with a heat source present. In Chapter 1 we havedescribed the BEM for

discretising the heat equation. With this method, the heat equation is first multiplied

by the fundamental solution and then integrated with the assistance of Green’s identity.

This leads to a boundary integral equation which can be discretised with resulting inte-

gral coefficients that can be evaluated analytically. The initial and boundary conditions

are also imposed.

In an inverse problem, additional conditions are required to determine uniquely the

unknown functions. However, this information has to come from measurements which

are contaminated with noise unavoidably. If the problem is ill-posed then small errors

177

Chapter 8. 178

in the measurement data result in highly and unbounded output solutions. Therefore,

regularisation methods need to be employed to deal with thisinstability.

In this thesis, many regularisation methods have been utilised together with the

BEM. A popular regularisation method, the Tikhonov regularisation, has been used

with orders zero, one and two. Additionally, for comparison, the TSVD method has

also been considered in Chapters 3 and 6. Moreover, in Chapter 4, the smoothing spline

technique has been considered as a regularisation method for seeking a regularised

first-order derivative of a noisy function.

Regularisation methods require a proper choice of the regularisation parameter.

There are many methods such as theL-curve method, the GCV criterion, and the

discrepancy principle which are all popular and successfulmethods for choosing the

regularisation parameter. TheL-curve method is the simplest method for choosing the

regularisation parameter. This method suggests choosing the parameter at the corner

of the L-curve which is a plot of the solution norm versus the corresponding resid-

ual for many positive regularisation parameters. Alternatively, we have also used the

GCV criterion in order to indicate a regularisation parameter, this method is based on

the minimising the GCV function of various positive regularisation parameters. When

the amount of noise is known, the discrepancy principle was proposed to be another

method for choosing the regularisation parameter. This method is more rigorous since

it requires the knowledge of the noise level with which the input data is contami-

nated. Furthermore, in Chapter 6, the selection of two regularisation parameters has

been based on theL-surface method, this method is a natural extension of theL-curve

method used for the selection of a single regularisation parameter. For comparison, the

simple trial and error technique has also been employed, i.e. various regularisation pa-

rameters were tested with gradually increasing value untiloscillations in the numerical

solutions have been stabilised.

To test the accuracy and stability of the BEM combined with regularisation meth-

ods, numerical examples consisting of various cases of unknown functions, such as

smooth continuous, non-smooth continuous and discontinuous functions, have been

Chapter 8. 179

illustrated and compared with their analytical solutions,where available. Otherwise,

in the cases where the analytical solution for the temperature is not available, we have

used the numerical solution of the corresponding direct problem and set the mesh size

to be different in the corresponding inverse problem presented in Chapters 3 and 5.

This is in order to avoid committing an inverse crime, see [32].

In summary, all numerical results with/without noise contamination have been

found to be accurate and stable. In Chapter 2, the determination of the time-dependent

heat source function and the temperature subjected to threegeneral boundary con-

ditions has been considered. These three conditions have been distinguished to be

six separate cases of boundary and overdetermination conditions and generating six

inverse problems. Some cases were found to be ill-conditioned; then the Tikhonov

regularisation with orders zero, one and two have been used on both exact and noisy

data. Whereas other cases were found to be well-conditionedand the use of BEM has

processed well for the inverse problem with no use of regularisation for the exact data,

but the regularisation was still needed when noise was present.

In Chapter 3, the identification of the time-dependent heat source and the temper-

ature subjected to a periodic boundary condition, a Robin boundary condition and an

integral overdetermination condition has been considered. The BEM has been devel-

oped and combined with two regularisation methods; the Tikhonov regularisation and

the TSVD method. A couple of benchmark test examples have been presented in order

to illustrate the accuracy of the numerical results. No regularisation was required in the

case of exact data and we found that the least-squares procedure and the SVD method

produced the same accurate numerical results. When noise was added, theL-curve

method and the discrepancy principle were selected for the appropriate choice of the

regularisation parameter, when using the Tikhonov regularisation of orders zero, one

and two, and the truncation level, when using the TSVD. Numerical results obtained

by using the BEM combined with either the TSVD or the zeroth-order Tikhonov regu-

larisation have been formed similar. The higher-order regularisation for smooth source

recovery gave more accurate results than the lower-order ones, while for non-smooth

Chapter 8. 180

sources the conclusion was reversed.

In Chapter 4, we have investigated the reconstruction of thetime-dependent blood

perfusion coefficient and the temperature in the bioheat equation subjected to the same

boundary and overdetermination conditions as in Chapter 3.A simple transformation

reduced the bioheat equation to be the classical heat equation, but now the overdeter-

mination condition contained the unknown source function.Two numerical examples

have been solved using the BEM. One example has been considered together with the

Tikhonov regularisation combined with the higher-order (of accuracy) finite difference

and use the GCV method as the choice of regularisation parameter. The second exam-

ple has used the BEM together with a smoothing spline technique for differentiating a

noisy function with apriori and aposteriorichoices of the regularisation parameters.

Chapter 5 presented an identification of the time-dependentheat source and the

temperature for the heat equation subjected to the non-classical boundary and integral

overdetermination conditions. We have utilised the same technique as before based

on the BEM together with the Tikhonov regularisation method. Three benchmark test

examples have been considered with smooth and non-smooth continuous source func-

tions to illustrate the accuracy and stability of the numerical results. Utilising the GCV

method as choice of regularisation parameter has performedwell to obtain stable and

accurate solution in all the investigated examples. We havealso found that there was

not much significant difference value of the regularisationparameter given by the dis-

crepancy principle or the trial and error technique.

In Chapter 6, we have investigated the more challenging identification of two un-

known source functions; the time- and space-dependent components of an additive

heat source and the temperature in the one-dimensional heatequation subjected to in-

terior point and time integral overdetermination conditions. The BEM was combined

with either the Tikhonov regularisation or the TSVD to solvethe inverse problem with

various selections of the regularisation parameter and thetruncation level based on the

L-curve method, the discrepancy principle and the GCV criterion when a single reg-

ularisation parameter was considered. We have also extended the analysis to the case

Chapter 8. 181

when two regularisation parameters were present and chose these parameters by using

theL-surface method.

The final case of inverse heat source problem presented in Chapter 7 was a non-

linear case study. This consisted of the simultaneous determination of multiplicative

space- and time-dependent source components and the temperature for the heat equa-

tion subject to homogeneous Neumann boundary condition, specified interior, and final

time temperature measurements. The numerical discretisation was based on the BEM

combined with a Tikhonov regularisation procedure. The resulting nonlinear optimi-

sation problem was solved using the MATLAB routinelsqnonlin. The hybrid-order

combination of the first- and second-order Tikhonov regularisation has achieved a sta-

ble and accurate numerical solution.

Throughout the thesis, the retrieved numerical results were found to be accurate and

stable concluding the reliability of the BEM combined with the various regularisation

techniques for solving a wide range of inverse source problems for the heat equation.

8.2 Future work

As we have studied so far, the use of the BEM combined with regularisation methods

can be developed for solving inverse source problems for theheat equation under var-

ious types of boundary and overdetermination conditions. This supports the idea that

the BEM combined with regularisation methods can also perform well in other related

possible future work, as follows.

(i) An inverse source problem related to that of Chapter 5 andgiven by the follow-

Chapter 8. 182

ing system of equations:

ut = uxx + r(t)f(x, t), (x, t) ∈ DT ,

u(x, 0) = u0(x), x ∈ [0, 1],

u(0, t) = 0, t ∈ (0, T ],

ut(1, t) + ux(1, t) + ϕ(u(1, t)) = 0, t ∈ (0, T ],

∫ 1

0u(x, t) dx = E(t), t ∈ [0, T ],

(8.1)

whereϕ is a given nonlinear function, has recently been investigated in [50] but with

no numerical study. It would be interesting to study the numerical reconstruction of

the time-depending heat sourcer(t) and the temperatureu(x, t) satisfying this inverse

problem by using the BEM together with the regularisation methods presented in this

thesis.

(ii) Another possible future work for the one-dimensional study is the combination

of the identifications in Chapters 6 and 7 for a more general heat source containing

both additive and multiplicative components, see [47]. This work is an identification

of finding the time-dependent source functionsr(t), r1(t), the space-dependent source

functionss(x), s1(x) and the temperatureu(x, t) which satisfy the heat equation

ut(x, t) = uxx(x, t) + r(t)s(x) + r1(t) + s1(x), (x, t) ∈ DT , (8.2)

subject to the initial condition (1.7), the homogeneous Neumann boundary condition

(7.3), and the additive measurements

u(X0, t) = χ(t), u(X1, t) = χ1(t), t ∈ [0, T ], (8.3)

u(x, T1) = β1(x), u(x, T2) = β2(x), x ∈ [0, L], (8.4)

at the fixed sensor locations0 ≤ X0 < X1 ≤ L and the fixed times0 < T1 < T2 ≤ T .

Chapter 8. 183

Fixing conditions are also required as

r(T1) = γ1, r(T2) = γ2, s(X0) = S0, s1(X0) = S0, (8.5)

where the functionsχ, χ1, µ1, µ2 and the constantsγ1, γ2, S0, S0 are given. This in-

verse problem is very challenging because it is nonlinear study and the Matlab routine

lsqnonlin will be required.

(iii) The multi-dimensional inverse source problem for theheat equation is also

very interesting to study further with the BEM. The following inverse source problem

can be further studied, see Cannon [8] and Yamamoto [62]. LetΩ be a bounded do-

main inRn, n = 1, 2, 3. Then, one can consider the inverse problem of finding the

temperatureu(x, t) for (x, t) ∈ Ω × (0, T ) and the space-dependent heat sourcef(x)

for x ∈ Ω, satisfying the transient heat conduction equation

∂u

∂t(x, t) = ∇2u(x, t) + r(t)f(x), (x, t) ∈ Ω× (0, T ), (8.6)

subject to the initial condition

u(x, 0) = u0(x), x ∈ Ω, (8.7)

and the overspecified Cauchy boundary data

u(x, t) = β(x, t), (x, t) ∈ ∂Ω× (0, T ), (8.8a)

∂u

∂n(x, t) = ϑ(x, t), (x, t) ∈ Γ× (0, T ), (8.8b)

or

∂u

∂n(x, t) = ϑ(x, t), (x, t) ∈ ∂Ω × (0, T ), (8.9a)

u(x, t) = β(x, t), (x, t) ∈ Γ× (0, T ), (8.9b)

Chapter 8. 184

whereΓ ⊂ ∂Ω is a non-empty open subset of the boundary∂Ω, andr, u0, β andϑ are

known functions.

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